From MAILER-DAEMON Sat Nov 24 09:20:37 2007 Date: 24 Nov 2007 09:20:37 -0400 From: Mail System Internal Data Subject: DON'T DELETE THIS MESSAGE -- FOLDER INTERNAL DATA Message-ID: <1195910437@mta.ca> X-IMAP: 1191260185 0000000106 Status: RO This text is part of the internal format of your mail folder, and is not a real message. It is created automatically by the mail system software. If deleted, important folder data will be lost, and it will be re-created with the data reset to initial values. From rrosebru@mta.ca Mon Oct 1 13:57:01 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 01 Oct 2007 13:57:01 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IcOUy-0005sA-8q for categories-list@mta.ca; Mon, 01 Oct 2007 13:52:08 -0300 Date: Mon, 1 Oct 2007 12:28:56 -0400 (EDT) From: Jeff Egger Subject: categories: Quantale Theory 101 [was: is 0 prime?] To: categories list MIME-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 1 --- Bill Lawvere wrote: > Indeed, as Jeff points out, we learned from Kummer and Dedekind to repl= ace > elements by ideals, but we categorists have been late in providing a cl= ear > account of this transition and, in particular, of the reason why the > result is not primarily a lattice, but a monoidal closed category with > colimits. In fact, I think that the process of moving from rings to lattices of ide= als=20 should be seen in two stages. The first stage is to observe that the fun= ctor=20 Ab ---> Sup which maps an abelian group to its lattice of subobjects come= s equipped with a natural monoidal structure. [Sup denotes the category of= =20 complete lattices and sup-homomorphisms.] Thus monoids in Ab (rings) get mapped to monoids in Sup (quantales). [Of course, one can replace Ab, no= t=20 just by CMon, but by other interesting categories, such as Ban.] The second stage is to pare the quantale of all (additive) subgroups of a= =20 ring down to that of ideals; but (left-, right-, two-sided) ideals are, b= y definition, precisely the (left-, right-, two-sided) elements of the=20 quantale of subgroups, so all that remains to do is properly describe thi= s process of paring an arbitrary quantale to its "subquantale" of two-sided elements (subquantale in the sense of sub-semigroup, not sub-monoid). =20 Restricting to the category of commutative quantales---which I shall adop= t=20 as the case of interest, for the purposes of the present discussion (sinc= e=20 it started out with the ring of integers)---we see that this functor is l= eft adjoint to the forgetful functor from the category of {commutative quanta= les whose unit is top}. [The problem with the general case is that the=20 "obvious" unit map: x |-> T&x&T (where T denotes top and & is quantale multiplication) need not be a quantale homomorphism; there appear to be=20 several ways of fixing this, and I do not yet know which is the best.] The nice thing about this approach is that one then recognises the second stage as leading naturally to a third: namely, collapsing down to the fra= me=20 of radical ideals (which is the topology of the space of primes I referre= d=20 to in my previous post). In particular, if one regards this third stage = as erroneous=20 > The distributive lattice of radical ideals is refined to the monoidal > poset of all ideals. then one should probably regard the second stage as equally erroneous ---which is the position that the quantale theory community has largely=20 agreed upon. =20 As to the question of "why?", I have a very biased and unscientific=20 answer: Sup is the most awesome category. Cheers, Jeff Egger. Ask a question on any topic and get answers from real people. Go to= Yahoo! Answers and share what you know at http://ca.answers.yahoo.com From rrosebru@mta.ca Mon Oct 1 13:57:01 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 01 Oct 2007 13:57:01 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IcOVY-0005yw-GC for categories-list@mta.ca; Mon, 01 Oct 2007 13:52:40 -0300 Date: Mon, 1 Oct 2007 12:28:39 -0400 (EDT) From: Jeff Egger Subject: categories: Ideal Theory 101 [was: is 0 prime?] To: categories list MIME-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 2 --- Vaughan Pratt wrote: > I don't know much about ring theory, so I > could be confused about this, but I would have thought intersecting the= m > could only get you the square-free ideals.=20 This is correct; there is simply no way of getting around the fact that=20 ideals form not just a lattice but carry a quantale structure derived=20 from the ring. [See my next post and the quotation below.] =20 --- Bill Lawvere wrote: > The ideal product under discussion is a key > ingredient in a construction of unions of subspaces that takes into > account the clashes.=20 --- Vaughan Pratt wrote: > Starting from the prime > power ideals takes care of that but what's the trick for getting all th= e > ideals from just the prime ideals? The category Div was my suggestion > for that, but if there's a more standard approach in ring theory I'd be > happy to use that instead (or at least be aware of it---Div is starting > to grow on me). I'd point you to Wikipedia, only the relevant articles are somewhat=20 scattered about. Briefly, every ideal in a Noetherian ring can be=20 written as a finite intersection of _primary_ ideals, and this can=20 be made essentially unique by adding appropriate restrictions.=20 To obtain a more easily recognisable version of the Fundamental=20 Theorem of Arithmetic, it then remains to determine under what=20 circumstances a primary ideal must be a prime power. [A good=20 counter-example is Z[x,y], where the ideal (x,y^2) is primary,=20 but falls strictly between the prime ideal (x,y) and its square (x,y)^2=3D(x^2,xy,y^2).] =20 A Noetherian integral domain which does have this extra property=20 is called a Dedekind domain; examples include the ring of algebraic=20 integers w.r.t. an arbitrary number field---proving the latter result=20 (which is connected to an infamously incorrect proof of Fermat's=20 last theorem) is commonly cited as Dedekind's original motivation=20 for defining ideals. See http://en.wikipedia.org/wiki/Primary_decomposition and http://en.wikipedia.org/wiki/Dedekind_domain for details. =20 Cheers, Jeff Egger. Ask a question on any topic and get answers from real people. Go to= Yahoo! Answers and share what you know at http://ca.answers.yahoo.com From rrosebru@mta.ca Tue Oct 2 13:27:41 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 02 Oct 2007 13:27:41 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IckPl-0001nd-U8 for categories-list@mta.ca; Tue, 02 Oct 2007 13:16:06 -0300 Date: Mon, 01 Oct 2007 23:40:59 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories list Subject: categories: Re: Quantale Theory 101 [was: is 0 prime?] Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 3 > As to the question of "why?", I have a very biased and unscientific > answer: Sup is the most awesome category. Oh, *there*'s the problem. I was getting quite puzzled about all this stuff. Presumably by Sup you mean what Peter Johnstone calls CSLat, complete semilattices, which is a lovely self-dual category. (If not ignore the following.) According it the status of "the most awesome" however is a symptom of not yet having come to grips with the joy of Chu, a more awesome self-dual category (fully) embedding CSLat in a duality-preserving and concrete-preserving way while exhibiting that duality as simply matrix transposition, yet still not *the* most awesome. And all that just in Chu(Set,2). Chu(Set,8) embeds Grp, and concretely at that, which is more awesome but still not awesome to the max. More awesome yet is that you can concretely embed every category of relational structures of total arity n in Chu(Set,2^n)---Grp fits that description on account of the group multiplication being a ternary relation, whence Chu(Set,8)). And so on. If going up only reduces the awe, then one should instead go down from CSLat for greater awe. God and the devil command a degree of awe that the middle class is hard pressed to match. Not only am I not a ring theorist but it's never occurred to me even to play one on the Internet. On the matter of the ideals of R, it would be very nice if they were just the endomorphisms of R but presumably that doesn't work on the ground that not every quotient of R embeds as a subring of R---if that's wrong then I'm really confused. I'm not a category theorist either but I do try. Isn't the obvious gadget to extract from R not its lattice of ideals but its category of quotients suitably defined? Bill, is that what you were getting at? Vaughan From rrosebru@mta.ca Tue Oct 2 13:27:42 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 02 Oct 2007 13:27:42 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IckRO-00020X-DM for categories-list@mta.ca; Tue, 02 Oct 2007 13:17:46 -0300 Date: Tue, 02 Oct 2007 01:37:45 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories list Subject: categories: On FOM, free Boolean algebras are semantic, not syntactic Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 4 [Note from moderator: Thanks Vaughan...] This is a short note to express my appreciation for Bob Rosebrugh's quiet but effective management of this list over a great many years (are we up to two decades yet, Bob?) When things have been going swimmingly long enough it becomes hard to picture what an alternative universe might have been like. But not impossible. A few days ago I signed up again for the Foundations of Mathematics mailing list after many years away from it, thinking that maybe it had improved since I left it. When someone posted a message claiming that well-formed formulas needed to be presented inductively and that this requirement was giving him great angst, and someone else responded with a pointer to a 61-page paper explaining how to define syntax using category theory, I responded with a contrarian post, illustrated with a short definition of wff using only sets, inclusion, functions, and linear orders, with no mention of induction. Central to the definition was the notion of wff as a function 2^P --> 2, recognizable as an element of the free Boolean algebra on P (here the set of predicates appearing in the wff) but without actually saying "free Boolean algebra." My post was rejected for submission on the ground that it was "deemed inappropriate by the moderator," with the further explanation that I was "confusing the entirely syntactic notion of formula with semantic notions." Either the (anonymous) moderator has never seen a representation of a free Boolean algebra, or views all algebras including the free ones as semantics and hence unfit for posting on FOM in connection with the definition of wff. A slightly more convincing ground for rejection might have been that 2^X --> 2 is the set of terms at X of the monad for Boolean algebras, and that anyone familiar with the Kleisli construction would see right away that I was just trying to disguise the associated inductive definition of wff by semantic smoke and mirrors. To which I would have responded first with Sol Feferman's question, "What rests on what?", and second with "It was you who picked the initial adjunction for that monad, how do you know I didn't have the final one in mind?" Had the anonymous moderator at least mentioned Kleisli we might have had a basis for debating the appropriateness of the rejection on such grounds, though it becomes unpleasant to have to spend more time defending a submission to a faceless moderator than writing it in the first place. Absent such I concluded that FOM had fallen into ignorant hands, making it little more than a time sink for MOPDAL21, Members Of the Project to Drag Archaic Logic into the 21st century. With such decisions, those of FOM's moderators still having a reputation to maintain would do well to keep their rejection messages anonymous. So again, thank you Bob for making this list an enjoyable free-for-all of ideas. Things would have been less fun if you'd decided that anything outside the scope of CTWM was heresy unfit for posting. And thank you for *never* telling me I'm confused. Even though you may have suspected it on many occasions, some even apparent to me. Vaughan Pratt From rrosebru@mta.ca Tue Oct 2 21:42:44 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 02 Oct 2007 21:42:44 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IcsDp-0001ON-FI for categories-list@mta.ca; Tue, 02 Oct 2007 21:36:17 -0300 Date: Tue, 2 Oct 2007 18:29:58 -0400 (EDT) From: Jeff Egger Subject: categories: Of chickens and eggs [was: is 0 prime?] To: categories list MIME-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 5 --- Vaughan Pratt wrote: > Presumably by Sup you mean what Peter Johnstone calls CSLat, > complete semilattices, which is a lovely self-dual category. =20 Yes, indeed I did provide an equivalent definition: > > [Sup denotes the category of complete lattices and sup-homomorphisms.= ]=20 > According it the status of "the most awesome" however is a symptom of > not yet having come to grips with the joy of Chu,=20 At the risk of appearing pretentious, I'd like to quote Chekhov: de gusti= bus, aut bene aut nihil. ;) [Incidentally, I do like Chu categories, but I will play devil's advocate here.] > a more awesome > self-dual category (fully) embedding CSLat in a duality-preserving and > concrete-preserving way=20 ...but not tensor-preserving? I could just as easily say that Chu(Set,2)= =20 is (equivalent to) a lluf subcategory of Rel^2 (2 here denoting the arrow category), which is in turn (equivalent to) a full subcategory of Sup^2;=20 the latter carries a fascinating *-autonomous structure derived from thos= e=20 of Sup and 2, and the composite embedding is duality-preserving (though o= nly=20 the first part is "concrete-preserving"). =20 > [...] which is more awesome but still not awesome to the max. =20 Word. =20 > If going up only reduces the awe, then one should instead go down from > CSLat for greater awe. =20 The trouble with (Dedekind-)infinite things is that one can argue about=20 which way is up and which way is down. For example, both the forgetful functor Sup ---> Pos, and its left adjoint can be regarded as "embeddings= " ---thus one could perversely regard complete (semi)lattices as more, not less, general than arbitrary posets. =20 > Not only am I not a ring theorist but it's never occurred to me even to > play one on the Internet.=20 I hope no-one would accuse me of "playing the ring theorist" on the=20 Internet or elsewhere, merely as a result of quoting some of the subject'= s most celebrated theorems. [I was glad to learn that I have forgotten a=20 smaller chunk of my undergraduate education than I would have suspected.]= =20 Cheers, Jeff. P.S. It has been pointed out to me, by a reader of this list, that the=20 "conventional wisdom" I quoted in re the history of ideal theory is=20 flawed (as I suspected, for no deeper reason than a profound mistrust=20 of conventional wisdom). > > [...] is commonly cited as Dedekind's original motivation=20 > > for defining ideals. >=20 > Hi Jeff, > in fact Kummer defined ideal numbers and proved the Fermat > conjecture for regular primes before Lame' presented the > fallacious argument (by some years, I think, but I can't recall > just how many). There's a lot of information about this in > the Edwards book named after the conjecture (and some more in > his recent book on constructive algebra). From rrosebru@mta.ca Tue Oct 2 21:42:44 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 02 Oct 2007 21:42:44 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IcsCw-0001HJ-2m for categories-list@mta.ca; Tue, 02 Oct 2007 21:35:22 -0300 Date: Tue, 02 Oct 2007 22:38:29 +0200 From: Joachim Kock Subject: categories: CRM Workshop on Derived Categories To: categories@mta.ca MIME-version: 1.0 Content-type: text/plain; charset=us-ascii Content-transfer-encoding: 7BIT Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 6 WORKSHOP ON DERIVED CATEGORIES CRM Barcelona November 5 to 14, 2007 Scientific organisers: Leovigildo Alonso Tarrio (Universidade de Santiago de Compostela) Ana Jeremias Lopez (Universidade de Santiago de Compostela) Amnon Neeman (Australian National University, Canberra) A specialised workshop on Derived Categories will be held at the CRM Barcelona from November 5 to 14, 2007, within a one-year research programme on current trends in Homotopy Theory, Category Theory, and related disciplines. The workshop has been planned so as to allow ample time for discussions and interaction with participants, with a few talks each day and open discussion sessions. The participation of young researchers is much encouraged. If you are interested in giving a talk or need financial support to participate, please contact any of the scientific organisers before October 15. Registration for the workshop can be made online at the address http://www.crm.cat/Derived_Categories/. For organisational inquiries, including lodging possibilities, please see the information given on the above website or contact the CRM Secretariat at Derived_Categories@crm.cat. From rrosebru@mta.ca Tue Oct 2 21:42:44 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 02 Oct 2007 21:42:44 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IcsCF-0001AD-KE for categories-list@mta.ca; Tue, 02 Oct 2007 21:34:39 -0300 Date: Tue, 2 Oct 2007 13:05:50 -0400 (EDT) From: Michael Barr To: categories list Subject: categories: Re: On FOM, free Boolean algebras are semantic, not syntactic Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 7 [Note from moderator: with apologies for intervening twice in one day, I'll forward Mike's note with thanks and the information that the list will be untended until Sunday. That the categories list is lively is a reflection of its many excellent contributors. The moderation consists almost entirely of deciding which conference announcements are of at least some interest. To be precise we've been operating for 10001 internet years - just getting going.] Let me second Vaughan's commendation of Bob. I don't think it has been two decades quite (maybe 15 years) but of all the mailing lists I am aware of this is the one that is most vigorous and useful. Most seem to have fallen into disuse and Vaughan reports on another one that seems to be too tightly moderated. I don't actually know if Bob really moderates them but this one is the best I have seen. Michael On Tue, 2 Oct 2007, Vaughan Pratt wrote: > > [Note from moderator: Thanks Vaughan...] > > This is a short note to express my appreciation for Bob Rosebrugh's > quiet but effective management of this list over a great many years (are > we up to two decades yet, Bob?) When things have been going swimmingly > long enough it becomes hard to picture what an alternative universe > might have been like. > > But not impossible. A few days ago I signed up again for the > Foundations of Mathematics mailing list after many years away from it, > thinking that maybe it had improved since I left it. > > When someone posted a message claiming that well-formed formulas needed > to be presented inductively and that this requirement was giving him > great angst, and someone else responded with a pointer to a 61-page > paper explaining how to define syntax using category theory, I responded > with a contrarian post, illustrated with a short definition of wff using > only sets, inclusion, functions, and linear orders, with no mention of > induction. Central to the definition was the notion of wff as a > function 2^P --> 2, recognizable as an element of the free Boolean > algebra on P (here the set of predicates appearing in the wff) but > without actually saying "free Boolean algebra." > > My post was rejected for submission on the ground that it was "deemed > inappropriate by the moderator," with the further explanation that I was > "confusing the entirely syntactic notion of formula with semantic notions." > > Either the (anonymous) moderator has never seen a representation of a > free Boolean algebra, or views all algebras including the free ones as > semantics and hence unfit for posting on FOM in connection with the > definition of wff. > > A slightly more convincing ground for rejection might have been that > 2^X --> 2 is the set of terms at X of the monad for Boolean algebras, > and that anyone familiar with the Kleisli construction would see right > away that I was just trying to disguise the associated inductive > definition of wff by semantic smoke and mirrors. To which I would have > responded first with Sol Feferman's question, "What rests on what?", and > second with "It was you who picked the initial adjunction for that > monad, how do you know I didn't have the final one in mind?" > > Had the anonymous moderator at least mentioned Kleisli we might have had > a basis for debating the appropriateness of the rejection on such > grounds, though it becomes unpleasant to have to spend more time > defending a submission to a faceless moderator than writing it in the > first place. > > Absent such I concluded that FOM had fallen into ignorant hands, making > it little more than a time sink for MOPDAL21, Members Of the Project to > Drag Archaic Logic into the 21st century. With such decisions, those of > FOM's moderators still having a reputation to maintain would do well to > keep their rejection messages anonymous. > > So again, thank you Bob for making this list an enjoyable free-for-all > of ideas. Things would have been less fun if you'd decided that > anything outside the scope of CTWM was heresy unfit for posting. > > And thank you for *never* telling me I'm confused. Even though you may > have suspected it on many occasions, some even apparent to me. > > Vaughan Pratt > > From rrosebru@mta.ca Wed Oct 3 09:40:48 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 03 Oct 2007 09:40:48 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Id3QE-0004Iz-IC for categories-list@mta.ca; Wed, 03 Oct 2007 09:33:50 -0300 Date: Wed, 03 Oct 2007 01:27:08 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories list Subject: categories: Re: Of chickens and eggs [was: is 0 prime?] Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 8 Jeff Egger wrote: > > ...but not tensor-preserving? Tensor-preserving is the exception, the rule is a tensorial strength, as in this case. Vaughan From rrosebru@mta.ca Sat Oct 6 23:19:16 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 06 Oct 2007 23:19:16 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IeLVf-00036S-Vs for categories-list@mta.ca; Sat, 06 Oct 2007 23:04:48 -0300 Date: Fri, 5 Oct 2007 08:52:29 -0400 (EDT) From: Michael Barr To: Categories list Subject: categories: Help! MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 9 What would you say to an undergraduate math club about categories? I have been thinking about it, but I am not sure what to say. Talk about cohomology, which is what motivated E-M? I don't think so. Talk about dual spaces of finite-dimensional vector spaces? Maybe, but then what? Michael From rrosebru@mta.ca Sat Oct 6 23:19:16 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 06 Oct 2007 23:19:16 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IeLXh-0003Bi-OJ for categories-list@mta.ca; Sat, 06 Oct 2007 23:06:53 -0300 Mime-Version: 1.0 (Apple Message framework v752.2) To: Categories From: JeanBenabou Subject: categories: "Historical terminology" Date: Fri, 5 Oct 2007 16:52:40 +0200 Content-Transfer-Encoding: 7bit Content-Type: text/plain;charset=US-ASCII;delsp=yes;format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 10 Dear colleagues I need your help for the following questions: (i) Who gave the name of "cartesian" to categories with finite limits? When was this name given? What is the first published paper where this name occurs? (ii) Same questions for "cartesian closed" (iii) Same questions again for "locally cartesian closed". Moreover, in this case, does the precise definition imply that such a category has a terminal object? Thanks for your help, Jean --Apple-Mail-1-687879601 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=ISO-8859-1 Dear colleagues=A0

I need your help for the = following questions:

(i) Who gave the name of = "cartesian"=A0=A0to categories with finite limits? When was this = name given? What is the first published paper where this name = occurs?
(ii) Same = questions for "cartesian closed"
(iii) Same questions again for "locally = cartesian closed". Moreover, in this case, does the precise = definition imply that such a category has a terminal = object?

Thanks for your = help,

Jean
= --Apple-Mail-1-687879601-- From rrosebru@mta.ca Sat Oct 6 23:19:16 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 06 Oct 2007 23:19:16 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IeLYh-0003E8-8v for categories-list@mta.ca; Sat, 06 Oct 2007 23:07:55 -0300 Date: Fri, 5 Oct 2007 12:10:07 -0400 (EDT) From: Jeff Egger Subject: categories: Re: Ideal Theory 101 [was: is 0 prime?] To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 11 I thought that I had explained my point of view clearly=20 enough, but apparently I haven't. =20 If A and B are (additive) subgroups of a ring R (commutative or otherwise), then A.B is the image of the composite=20 A @ B ---> R @ R --m-> R (where @ denotes tensor product of abelian groups, and=20 m is the multiplication of the R, regarded as an arrow=20 in AbGp). What is mysterious about this? =20 We have a functor AbGp ---> Pos which maps an abelian group X to its set of subgroups; this uses only the existence of=20 an appropriate factorisation system on AbGp. It is, in fact,=20 also a monoidal functor with Sub(X) x Sub(Y) ---> Sub(X @ Y)=20 defined by (A,B) |-> (the image of) A @ B ---> X @ Y. Now regarding a ring as a monoidal functor 1 ---> AbGp,=20 we obtain a composite monoidal functor 1 ---> Pos, which=20 is a monoidal poset. Specifically, the multiplication=20 on Sub(R) is defined by=20 Sub(R) x Sub(R) ---> Sub(R @ R) --Sub(m)--> Sub(R) which is exactly what I described earlier. =20 The mystery, if there is one, is why this monoidal poset happens to be closed. My explanation is that the monoidal functor AbGp ---> Pos factors (as a monoidal functor) through the monoidal forgetful functor Sup ---> Pos. This can be=20 easily derived from the fact that AbGp is cocomplete and @=20 cocontinuous in each variable; in fact, weaker hypotheses=20 would seem to suffice. =20 Thus, again regarding a ring as a monoidal functor 1--->AbGp, we can consider the composite monoidal functor 1--->Sup; which=20 is nothing more nor less than a monoidal closed poset that=20 happens to be (co)complete---and its underlying monoidal poset=20 (i.e., the composite 1--->Sup--->Pos) is Sub(R) by definition=20 ... mystery solved! Not quite, I hear you say: we want ideals, not arbitrary=20 additive subgroups---but Sub(R) retains enough information=20 of R to remember which of its elements are ideals and which=20 are not: an ideal is an additive subgroup A such that=20 T.A=3DT=3DA.T, where T denotes the top element of Sub(R)---namely, R itself. This is the "second stage" referred to in my=20 "Quantale Theory 101" post, which is the process of turning a quantale into an "affine" quantale (one whose top element=20 is also its (multiplicative) unit). In the commutative case, at least, this poses no problem whatsoever. I hope this clarifies my point of view on the Kummer functor. I don't see its existence as having anything in particular to=20 do with commutative rings, but rather with those properties of=20 AbGp which cause the monoidal functor AbGp--->Pos to=20 a) exist, and b) factor through the monoidal forgetful functor Sup--->Pos ---which, as I sketched above, are not particularly rare ones. The rest is taken care of by a purely quantale-theoretic process. [But my comment about Sup being an awesome category had more to do with why the Kummer functor "should be" interesting (aside from=20 the obvious concrete considerations), rather than why it exists.] I admit that this isn't entirely satisfying if you really are=20 interested in ideals as representing quotients of a ring; but=20 I do think that it is a valid perspective, nevertheless, and=20 welcome further discussion on the topic. Cheers, Jeff. --- wlawvere@buffalo.edu wrote: > The awesome nature of Sup cannot be the reason why > the Kummer functor exists, since it is merely used for=20 > recording the result. The functor is "caused" rather=20 > by an internal feature of the domain category C of=20 > commutative rings: The category of quotient objects > of any given R has a binary operation * that is neither > sup nor inf even though in principle it can be=20 > expressed as a combination of limits and colimits. > We can call it R/ab=3DR/a *R/b but how does the=20 > operation * specialize to C concretely ? >=20 > Bill Get a sneak peak at messages with a handy reading pane with All new= Yahoo! Mail: http://mrd.mail.yahoo.com/try_beta?.intl=3Dca From rrosebru@mta.ca Sat Oct 6 23:19:16 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 06 Oct 2007 23:19:16 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IeLWl-000393-0u for categories-list@mta.ca; Sat, 06 Oct 2007 23:05:55 -0300 Date: Fri, 5 Oct 2007 13:18:52 GMT From: Oege.de.Moor@comlab.ox.ac.uk To: Subject: categories: PEPM 2008: abstracts due Oct 12 Content-Type: text/plain Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 12 >>> LAST CALL <<< >>> abstracts - Oct 12, full papers - Oct 17 <<< PEPM 2008 ACM SIGPLAN Workshop on Partial Evaluation and Program Manipulation January 7-8, 2008, San Francisco Keynotes by Ras Bodik (Berkeley) and Monica Lam (Stanford) Co-located with POPL http://www.program-transformation.org/PEPM08/WebHome PEPM is a leading venue for the presentation of cutting-edge research in program analysis, program generation and program transformation. Its proceedings are published by ACM Press; full details of the scope, submission process, and program committee can be found at the above URL. The program committee would particularly welcome submissions from category theorists on any topic relating to categorical justification of program fusion rules Abstracts are due on October 12, and the deadline for full paper submission is October 17. Prospective authors are welcome to contact the program chairs, Robert Glueck (glueck@acm.org) and Oege de Moor (oege@comlab.ox.ac.uk) with any queries they might have. From rrosebru@mta.ca Sat Oct 6 23:19:16 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 06 Oct 2007 23:19:16 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IeLUw-00034C-8q for categories-list@mta.ca; Sat, 06 Oct 2007 23:04:02 -0300 Subject: categories: Re: Ideal Theory 101 [was: is 0 prime?] Date: Thu, 04 Oct 2007 20:47:32 -0400 From: wlawvere@buffalo.edu To: categories list MIME-Version: 1.0 Content-Type: text/plain Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 13 The awesome nature of Sup cannot be the reason why the Kummer functor exists, since it is merely used for=20 recording the result. The functor is "caused" rather=20 by an internal feature of the domain category C of=20 commutative rings: The category of quotient objects of any given R has a binary operation * that is neither sup nor inf even though in principle it can be=20 expressed as a combination of limits and colimits. We can call it R/ab=3DR/a *R/b but how does the=20 operation * specialize to C concretely ? Bill Quoting Jeff Egger : > --- Vaughan Pratt wrote: > > I don't know much about ring theory, so I > > could be confused about this, but I would have thought intersecting > the> m > > could only get you the square-free ideals.=20 > > This is correct; there is simply no way of getting around the fact > that=20 > ideals form not just a lattice but carry a quantale structure > derived=20 > from the ring. [See my next post and the quotation below.] =20 > > --- Bill Lawvere wrote: > > The ideal product under discussion is a key > > ingredient in a construction of unions of subspaces that takes > into > > account the clashes.=20 > > --- Vaughan Pratt wrote: > > Starting from the prime > > power ideals takes care of that but what's the trick for getting > all th> e > > ideals from just the prime ideals? The category Div was my > suggestion > > for that, but if there's a more standard approach in ring theory > I'd be > > happy to use that instead (or at least be aware of it---Div is > starting > > to grow on me). > > I'd point you to Wikipedia, only the relevant articles are > somewhat=20 > scattered about. Briefly, every ideal in a Noetherian ring can > be=20 > written as a finite intersection of _primary_ ideals, and this > can=20 > be made essentially unique by adding appropriate restrictions.=20 > > To obtain a more easily recognisable version of the Fundamental=20 > Theorem of Arithmetic, it then remains to determine under what=20 > circumstances a primary ideal must be a prime power. [A good=20 > counter-example is Z[x,y], where the ideal (x,y^2) is primary,=20 > but falls strictly between the prime ideal (x,y) and its square > (x,y)^2=3D(x^2,xy,y^2).] =20 > > A Noetherian integral domain which does have this extra property=20 > is called a Dedekind domain; examples include the ring of > algebraic=20 > integers w.r.t. an arbitrary number field---proving the latter > result=20 > (which is connected to an infamously incorrect proof of Fermat's=20 > last theorem) is commonly cited as Dedekind's original motivation=20 > for defining ideals. > > See http://en.wikipedia.org/wiki/Primary_decomposition > and http://en.wikipedia.org/wiki/Dedekind_domain > for details. =20 > > Cheers, > Jeff Egger. > > > > > Ask a question on any topic and get answers from real people. > Go to> Yahoo! Answers and share what you know at > http://ca.answers.yahoo.com > > > >=20 From rrosebru@mta.ca Sat Oct 6 23:19:16 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 06 Oct 2007 23:19:16 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IeLaP-0003I0-CF for categories-list@mta.ca; Sat, 06 Oct 2007 23:09:41 -0300 Content-Disposition: inline Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset="iso-8859-1" MIME-Version: 1.0 From: Subject: categories: Re: Ideal Theory 101 [was: is 0 prime?] Date: Fri, 05 Oct 2007 13:59:00 -0400 To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 14 As I emphasized under (2) in my Sep29 posting,=20 the point of view or perspective on the Kummer=20 functor that factors it through the large category=20 of module categories is quite interesting and useful and thoroughly understood by categorists, and so hides no "mysteries" of a general nature. Jeff has=20 reiterated that point now in elegant detail.=20 But my point was that another perspective, at least=20 as important and at least as old, is perhaps not=20 yet so well explained categorically. Categories of spaces=20 are often analyzed in terms of algebras of functions,=20 hence subspaces in terms of epimorphisms of algebras, (localizations for open subspaces and) regular=20 epimorphisms for closed subspaces. Of course the corresponding congruence relations can sometimes be identified with ideals in some sense. But algebras may be something different from commutative rings, in particular there may be no (known) categories of "modules" in which they can be=20 identified with monoids (an important example is=20 Cinfinity spaces and algebras). Yet the concrete example of the category of commutative rings should give clues toward understanding the geometric phenomenon that another operation besides the lattice ones crops=20 up naturally on the closed subspaces in all these categories. The understanding sought is thus primarily about these=20 categories themselves. (The example contrasting a relief map with a flat paper one was mentioned to show that these infinitesimals are real.) "All these categories " includes many algebraic categories, but not all. For example in the simplest algebraic category (no operations), the only "resolution" of the contradiction between intersection and image is surely trivial ? On Fri Oct 5 12:10 , Jeff Egger sent: >I thought that I had explained my point of view clearly=20 >enough, but apparently I haven't.=20=20 > >If A and B are (additive) subgroups of a ring R (commutative >or otherwise), then A.B is the image of the composite=20 > A @ B ---> R @ R --m-> R >(where @ denotes tensor product of abelian groups, and=20 >m is the multiplication of the R, regarded as an arrow=20 >in AbGp). What is mysterious about this?=20=20 > >We have a functor AbGp ---> Pos which maps an abelian group >X to its set of subgroups; this uses only the existence of=20 >an appropriate factorisation system on AbGp. It is, in fact,=20 >also a monoidal functor with Sub(X) x Sub(Y) ---> Sub(X @ Y)=20 >defined by (A,B) |-> (the image of) A @ B ---> X @ Y. > >Now regarding a ring as a monoidal functor 1 ---> AbGp,=20 >we obtain a composite monoidal functor 1 ---> Pos, which=20 >is a monoidal poset. Specifically, the multiplication=20 >on Sub(R) is defined by=20 > Sub(R) x Sub(R) ---> Sub(R @ R) --Sub(m)--> Sub(R) >which is exactly what I described earlier.=20=20 > >The mystery, if there is one, is why this monoidal poset >happens to be closed. My explanation is that the monoidal >functor AbGp ---> Pos factors (as a monoidal functor) through >the monoidal forgetful functor Sup ---> Pos. This can be=20 >easily derived from the fact that AbGp is cocomplete and @=20 >cocontinuous in each variable; in fact, weaker hypotheses=20 >would seem to suffice.=20=20 > >Thus, again regarding a ring as a monoidal functor 1--->AbGp, >we can consider the composite monoidal functor 1--->Sup; which=20 >is nothing more nor less than a monoidal closed poset that=20 >happens to be (co)complete---and its underlying monoidal poset=20 >(i.e., the composite 1--->Sup--->Pos) is Sub(R) by definition=20 >... mystery solved! > >Not quite, I hear you say: we want ideals, not arbitrary=20 >additive subgroups---but Sub(R) retains enough information=20 >of R to remember which of its elements are ideals and which=20 >are not: an ideal is an additive subgroup A such that=20 >T.A=3DT=3DA.T, where T denotes the top element of Sub(R)---namely, >R itself. This is the "second stage" referred to in my=20 >"Quantale Theory 101" post, which is the process of turning >a quantale into an "affine" quantale (one whose top element=20 >is also its (multiplicative) unit). In the commutative case, >at least, this poses no problem whatsoever. > >I hope this clarifies my point of view on the Kummer functor. >I don't see its existence as having anything in particular to=20 >do with commutative rings, but rather with those properties of=20 >AbGp which cause the monoidal functor AbGp--->Pos to=20 > a) exist, and > b) factor through the monoidal forgetful functor Sup--->Pos >---which, as I sketched above, are not particularly rare ones. >The rest is taken care of by a purely quantale-theoretic process. >[But my comment about Sup being an awesome category had more to do >with why the Kummer functor "should be" interesting (aside from=20 >the obvious concrete considerations), rather than why it exists.] > >I admit that this isn't entirely satisfying if you really are=20 >interested in ideals as representing quotients of a ring; but=20 >I do think that it is a valid perspective, nevertheless, and=20 >welcome further discussion on the topic. > >Cheers, >Jeff. > >--- wlawvere@buffalo.edu wrote: > >> The awesome nature of Sup cannot be the reason why >> the Kummer functor exists, since it is merely used for=20 >> recording the result. The functor is "caused" rather=20 >> by an internal feature of the domain category C of=20 >> commutative rings: The category of quotient objects >> of any given R has a binary operation * that is neither >> sup nor inf even though in principle it can be=20 >> expressed as a combination of limits and colimits. >> We can call it R/ab=3DR/a *R/b but how does the=20 >> operation * specialize to C concretely ? >>=20 >> Bill > > > > Get a sneak peak at messages with a handy reading pane with All new = Yahoo! Mail: http://mrd.mail.yahoo.com/try_beta\?.intl=3Dca > > From rrosebru@mta.ca Sun Oct 7 10:53:14 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 07 Oct 2007 10:53:14 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IeWUo-0000iP-SA for categories-list@mta.ca; Sun, 07 Oct 2007 10:48:38 -0300 From: "George Janelidze" To: "Categories list" Subject: categories: Re: Help! Date: Sun, 7 Oct 2007 10:09:19 +0200 MIME-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 15 Dear Michael, The question is so impossibly big, and it was asked and answered in various form so many times, and no responsibility for the originality/completeness of the answer can be assumed... So, I am not afraid to begin with a few obvious remarks, looking forward to seeing many other remarks from others: 1. Many people believe that mathematics is about mathematical structures, but what is a mathematical structure in general? According to Bourbaki, one should begin with two finite sequences of sets, say, A, B, C,... and X, Y, Z,...; let us call them constant sets and variable sets respectively. Then build any scale, which is a finite sequence of sets obtained by taking finite products and power sets of the sets above. Then, (briefly and roughly!) call a structure (of a fixed type) an element of one of the sets in the scale satisfying certain conditions. For example: (a) a topological space (defined via open sets) has no constant sets, one variable set X, and its structure is an element t in PP(X) that is closed under finite intersections and arbitrary unions. (b) a vector space has one constant set A ("the set of scalars"), one variable set X ("the set of vectors"), and its structure can be defined as element (a,b,c,d) of P(AxAxA)xP(AxAxA)xP(AxXxX)P(XxXxX), where a, b, c, d are addition of scalars, multiplication of scalars, scalar (-on-vector) multiplication, and addition of vectors respectively; that (a,b,c,d) should satisfy familiar conditions of course. Then, according to Bourbaki again, structures are useless without morphisms - but what is a morphism? It turns out that only isomorphisms can be defined, and the class of morphisms should in each case be CHOSEN depending on the "experience" of working with a given class of structures in such a way, that it is closed under composition and contains all isomorphisms (or, better, also determine isomorphisms as invertible morphisms). Is it possible that the most fundamental concept of mathematics is described as such a monster? Is not it better to study abstract categories? And what is the problem of defining morphisms? To answer this question one should learn about functors, covariant and contravariant ones, and what do they preserve and what not. 2. Set theory is wonderful: it gives precise mathematical definitions to concepts that were only intuitively understood before. But... it often makes definitions complicated. And category theory very often solves this problem by using universal properties. For example the set N - better to say, the structure (N,0,s) - of natural numbers is "designed to count"; therefore N should have the first element 0 (unfortunately 0 is better than 1) and the successor function s from N to N - and be "the best such", i.e. initial such. Moreover, developing basic properties of this structure out of initiality is much easier than out of Peano axioms. In fact all classical number systems have simple elegant definitions via universal properties. Moreover, the axioms of set theory itself are much less elegant than their elementary-topos-theoretic counterparts. 3. We can say that set theory is more fundamental than arithmetics: e.g. children learn addition by counting the number of elements in the disjoint union. But category theory is more fundamental than set theory: e.g. it makes the disjoint union a more natural operation... but, more importantly (a) we all know that, say, a+b=b+a, ab=ba, PvQ<=>QvP,... for all natural numbers a and b and all logical formulas P and Q - but one needs category theory to see these as the same result (and we can add cartesian products, free product, intersection, union, and many other operations to it). (b) or, say, we all know that composites of injections are injections and composites of surjections are surjections - but again, one needs category theory to see these as the same result; (c) and even exponentiation and implication are the same... and categorical logic follows... 4. Linear algebra tells us that instead of working with linear transformations of finite-dimensional vector spaces we can work with matrices, but one cannot formulate this properly without using the concept of equivalence of categories (the category of finite-dimensional K-vector spaces is equivalent to the category of natural numbers with matrices with entries from K as morphisms). And there are so many other equivalences and dualities (that are not isomorphisms) playing fundamental roles in various branches of mathematics. Not to mention that the aforementioned matrices themselves arise from a categorical observation (finite products = finite coproducts). 5. Proper understanding of Eilenberg - Mac Lane work, and the work of their followers, friends, and not-quite-friends in category theory and proper understanding of what the 21st century mathematics would be without it obviously requires far better knowledge of mathematics than the 21st century students have. But may be we should at least say that our Fields Medal (Grothendieck) is not less than any other Fields Medal... George Janelidze ----- Original Message ----- From: "Michael Barr" To: "Categories list" Sent: Friday, October 05, 2007 2:52 PM Subject: categories: Help! > What would you say to an undergraduate math club about categories? I have > been thinking about it, but I am not sure what to say. Talk about > cohomology, which is what motivated E-M? I don't think so. Talk about > dual spaces of finite-dimensional vector spaces? Maybe, but then what? > > Michael From rrosebru@mta.ca Sun Oct 7 10:53:14 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 07 Oct 2007 10:53:14 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IeWTq-0000fF-Fk for categories-list@mta.ca; Sun, 07 Oct 2007 10:47:38 -0300 Mime-Version: 1.0 (Apple Message framework v752.2) To: Categories From: JeanBenabou Subject: categories: Re: "Historical terminology" Date: Sun, 7 Oct 2007 09:48:02 +0200 Content-Transfer-Encoding: 7bit Content-Type: text/plain;charset=US-ASCII;delsp=yes;format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 16 Cher Fred, Merci pour ta reponse rapide. Although your french is perfect, I shall continue in english, for the persons who are less familiar with french. (i) Your "guess" about cartesian closed categories is most certainly correct. I knew that Eilenberg/Kelly had explicitly used this name in their La Jolla paper, and it is probably the first instance, because "closed", in this sense, was first introduced in that paper, as far as I know.. (ii) Your "guess" about cartesian is not correct. Neither in Tohoku, nor in much later papers of his or any of his students, and also by me, was cartesian used in the sense of category with finite limits. If Grothendieck had used this name, which he has not, my "guess" is that he would have called cartesian categories with pull backs , because he and his students used the name "cartesian square" for square which is a pull back. Moreover this is special case of his notion of cartesian map in a fibration. (iii) I agree with you on the idea that the "natural" definition of locally cartesian closed category should not imply the existence of a terminal object. If I asked the question, it is because in Johnstone's "Elephant" he does assume a terminal object. Has such an assumption become, now, commonly accepted in the definition ? Thanks again, to you of course, and to whoever will help me to clarify (ii) and (iii) Jean > Salut, Jean, > > Without references at hand to consult, other than my failing > memory, I venture to hazard the following GUESSES at answers: > >> (i) Who gave the name of "cartesian" to categories with finite >> limits? When was this name given? What is the first published paper >> where this name occurs? > > This name I thought either you, or perhaps earlier Grothendieck, > had coined. When? Where? no idea (but if Grothendieck, then Tohoku?). > >> (ii) Same questions for "cartesian closed" > > My unverified guess: Eilenberg/Kelly, La Jolla, 1965. > >> (iii) Same questions again for "locally cartesian closed". > > No idea, but rather much later. > >> ... Moreover, >> in this case, does the precise definition imply that such a category >> has a terminal object? > > Here I have no answer at all, sorry, beyond this: IF the > definition of LCC is just that each "slice" category (but > not necessarily the category itself) be cartesian closed, > then most probably NOT. > >> Thanks for your help, > > I can only hope you find my guesses WERE actually of any help. > I fear, though, that they probably weren't at all. I'd be very > interested in learning the outcome of your survey, however. > >> Jean > > Cheers, > > -- Fred > From rrosebru@mta.ca Sun Oct 7 10:53:15 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 07 Oct 2007 10:53:15 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IeWVw-0000lM-Jk for categories-list@mta.ca; Sun, 07 Oct 2007 10:49:48 -0300 Message-ID: <010f01c808c3$b2dc8320$4601a8c0@RONNIENEW> From: "Ronnie Brown" To: "Categories list" Subject: categories: Re: Help! Date: Sun, 7 Oct 2007 10:23:18 +0100 MIME-Version: 1.0 Content-Type: text/plain;format=flowed;charset="iso-8859-1";reply-type=original Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 17 The great thing about categories is that they allow analogies between different mathematical structures: see the paper R. Brown and T. Porter) `Category Theory: an abstract setting for analogy and comparison', In: What is Category Theory? Advanced Studies in Mathematics and Logic, Polimetrica Publisher, Italy, (2006) 257-274. An example of the analogy is between the category of sets and the category of directed graphs: ``Higher order symmetry of graphs'', {\em Bull. Irish Math. Soc.} 32 (1994) 46-59. Here one easily sees non Boolean logics, of course. The word `analogy' seems to be underused in teaching undergraduates, but that is what abstraction is about, is it not? A teacher told me after a lecture on knots that was the first time he had heard the word analogy used in relation to mathematics! ( I discussed prime knots.) The other possibility is to advertise categorical structures: I advertised higher dimensional algebra to an international conference of neuroscientists in Delhi in 2003, pointing out the unlikelihood of the brain working entirely serially, and also the concept of colimit with an email analogy. A senior Indian neuroscientist came up to me afterwards and said that was the first time he had heard a seminar by a mathematician which made any sense! This is written up in (R. Brown and T. Porter), `Category theory and higher dimensional algebra: potential descriptive tools in neuroscience', Proceedings of the International Conference on Theoretical Neurobiology, Delhi, February 2003, edited by Nandini Singh, National Brain Research Centre, Conference Proceedings 1 (2003) 80-92. These are all downloadable from http://www.bangor.ac.uk/r.brown/publicfull.htm or my home page. See also http://www.bangor.ac.uk/r.brown/outofline/out-home.html for a general talk. As said before, I see higher dimensional algebra as the study of mathematical structures with operations defined under geometrical conditions, thus allowing a combination of algebra and geometry, in a way which even Atiyah might like (see his paper on `20th century mathematics' Bull LMS 44 (2002) 1-15, in which the words `category' and `groupoid' do not appear). I have found giving general talks makes one think hard about the underlying ideas and motivation. Ronnie ----- Original Message ----- From: "Michael Barr" To: "Categories list" Sent: Friday, October 05, 2007 1:52 PM Subject: categories: Help! > What would you say to an undergraduate math club about categories? I have > been thinking about it, but I am not sure what to say. Talk about > cohomology, which is what motivated E-M? I don't think so. Talk about > dual spaces of finite-dimensional vector spaces? Maybe, but then what? > > Michael From rrosebru@mta.ca Sun Oct 7 10:53:15 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 07 Oct 2007 10:53:15 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IeWXd-0000ob-CI for categories-list@mta.ca; Sun, 07 Oct 2007 10:51:33 -0300 From: "Marta Bunge" To: categories@mta.ca Subject: categories: RE: Help! Date: Sun, 07 Oct 2007 06:22:25 -0400 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 18 Hi, Michael, I am in the same predicament but, since I am speaking at this math club one week after you (November 6), I do hope to be able to use anything you do in your own talk! I also thought a lot about this problem and discarded one topic after another. Finally, I have decided to speak about the uses of infinitesimals in the synthetic calculus of variations, aiming at giving an algebraic (synthetic) proof of the well known fact that, for a paths functional ("energy"), its critical points agree with the geodesics. This requires that I introduce adjoint functors and cartesian closed categories and the notion of a ring object of line type. If you will do any of these yourself I could use it. Informally, I will argue constructively and acually prove things. Historical considerations may be briefly mentioned at the beggining of the talk, and the conceptual advantages of the synthetic method at the end. This will be an expanded portion of my paper "Synthetic Calculus of Variations" (with M. Heggie) in Contemporary Mathematics 30, 1983. I hope that this helps you as well as me. Best wishes, Marta >From: Michael Barr >To: Categories list >Subject: categories: Help! >Date: Fri, 5 Oct 2007 08:52:29 -0400 (EDT) > >What would you say to an undergraduate math club about categories? I have >been thinking about it, but I am not sure what to say. Talk about >cohomology, which is what motivated E-M? I don't think so. Talk about >dual spaces of finite-dimensional vector spaces? Maybe, but then what? > >Michael > > > _________________________________________________________________ Send a smile, make someone laugh, have some fun! Check out freemessengeremoticons.ca From rrosebru@mta.ca Mon Oct 8 10:13:57 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Oct 2007 10:13:57 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IesCn-0003Be-9w for categories-list@mta.ca; Mon, 08 Oct 2007 09:59:29 -0300 Date: Sun, 07 Oct 2007 09:16:49 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: Categories list Subject: categories: Re: Help! Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 19 George Janelidze wrote: > (a) a topological space (defined via open sets) has no constant sets, one > variable set X, and its structure is an element t in PP(X) that is closed > under finite intersections and arbitrary unions. The point of categories being (presumably) to shift the burden of structure from the objects to the morphisms, one would illustrate this point using your example by pointing out that the topological structure imputed to a space by the above definition is at least as well imputed with the definition of a space as the set of continuous functions to the space from the one-point space together with the set of continuous functions from it to the Sierpinski space. It sounded like you were headed in roughly that direction but then moved on to other points before getting there. > (b) a vector space has one constant set A ("the set of scalars"), one > variable set X ("the set of vectors"), and its structure can be defined as > element (a,b,c,d) of P(AxAxA)xP(AxAxA)xP(AxXxX)P(XxXxX), where a, b, c, d > are addition of scalars, multiplication of scalars, scalar (-on-vector) > multiplication, and addition of vectors respectively; that (a,b,c,d) should > satisfy familiar conditions of course. Ditto with the one-dimensional space in place of the one-point and Sierpinski space (which itself is a kind of one-dimensional space for topology). > 4. Linear algebra tells us that instead of working with linear > transformations of finite-dimensional vector spaces we can work with > matrices, but one cannot formulate this properly without using the concept > of equivalence of categories (the category of finite-dimensional K-vector > spaces is equivalent to the category of natural numbers with matrices with > entries from K as morphisms). You may be setting the bar for "proper" higher than necessary to satisfy us engineers. I'm currently involved in a computer project where the question of the proper formulation of matrices came up. One team had formulated them in terms of the Kleisli construction for monads as defined in CTWM, the monad in question being the one that you yourself would surely come up with for the variety Vct_C, C the complex numbers. Unfortunately that formulation was giving the computer conniptions. This could have been construed as bearing out your point were it not for the fact that another team came along with a reformulation of monads that overcame the difficulty. Since I know this list is good at keeping secrets (such as the secret of categories) I'll be happy to share with you all, in my next message, my confidential report on the current status of this reformulation. Our CEO is not convinced of the correctness of the reformulation, the fact that it fixed the buggy behavior notwithstanding, and has asked me for a qualified expert second opinion---where better than this list for a question about monads? Vaughan From rrosebru@mta.ca Mon Oct 8 10:13:57 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Oct 2007 10:13:57 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IesF3-0003Kz-J0 for categories-list@mta.ca; Mon, 08 Oct 2007 10:01:49 -0300 Date: Sun, 07 Oct 2007 09:44:52 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories list Subject: categories: Does Bind beat Kleisli in Hilbert space? Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 20 Project: Operation QCvac Sensitivity level: Black hole Situation report: Unanticipated overflow exception in a monad Reporting analyst: Vaughan Pratt Project status: On hold pending resolution Action item: Solicit qualified expert opinion Date: October 7, 2007 Situation summary. We're working on a quantum computer in anticipation of a Request for Proposal (RFP) for the next One Laptop Per Child (OLPC) computer for a value of "next" that is acceptable to our venture capitalists (VCs) yet feasible for our engineers. To be sure of not being out-competed we've assembled a crack team of physicists from Fermilab to get the physics right, category theorists from Fairfield, Iowa to design the linear algebra implementation, electrical engineers (EEs) from Silicon Valley to build the machine, Haskell programmers from Glasgow to implement the ideal third-party value-add software environment, and marketers from Boston to understand the market's needs and tastes. Marketing feels we have to be able to offer lots of storage (qubits). The physicists said no problem, they work in separable (countably dimensioned) Hilbert space all the time. (You see how physics works---physics is scale-invariant, what's good for describing the universe is good for describing computers.) Marketing said great, countably infinite storage will make us unbeatable, even Google will want one. Marketing wants to pitch the reliability of our machine. The category theorists said no problem, on their previous consulting job, D-Wave's 16-qubit quantum computer, they'd represented linear algebra as God's own monad, a monoid object (T,mu,eta) of Set^Set implementing matrix multiplication via the Kleisli construction. They took the functor T(X) to be the set C^X of X-dimensional almost-everywhere-zero complex vectors with T(f:X->Y): C^X --> C^Y sending v: C^X to the vector u: C^Y describable as starting with u = 0 and adding v_x to u_{f(x)} for all x in X, the multiplication mu_X: C^(C^X) --> C^X to send V: C^(C^X) to u: C^X with u_x the sum of V_v * v_x over all v in C^X (dot product), and the unit eta_X: X --> C^X as eta_X(x)(y) = (x=y) (the unit vectors). It worked like a charm. (You see how category theory works in computers---everything is an adjunction, or was in the 1970s, nowadays it's all done with monads.) The EEs expressed concern about the ambitious number of qubits. The category theorists reassured them that infinity held no fears for them as the infinite set C^(2^16) had worked fine in the D-Wave machine since mu only encountered finitely many nonzero values when summing over the domain C^(2^16) of T(f): C^(C(2^16)) --> C(2^16). The physicists reassured them that linear algebra lifted reliably to infinite dimensions provided the vectors were kept square summable as confirmed by a vast body of experimental evidence, so even though the sums would now be infinite they would still converge. (You see how systems analysis works---if monads and square-summability each coordinate well with the world they must coordinate well with one another.) Thus reassured, marketing said God's monad it is. Missionaries and the bible belt will buy millions. (You see how marketing works---logically this vision would have missionaries spending more money than Google, clearly absurd whence logic is false.) The EEs designed and built the first prototype in six days. The EEs were still in the lab on the seventh day because the machine was giving trouble. It was taking overflow exceptions in the course of performing output. By Tuesday the EEs had figured out what was going wrong. Their specification of the output module had it taking a system state v of dimension N (the natural numbers) and applying a projection p: C^N --> C^E collapsing v to one of a finite set E of eigenvectors whose eigenvalues constitute the possible outcomes of the measurement. They used the Kleisli implementation of application, quantumly realizing v and p as the respective functions v: 1 --> C^N and p: N --> C^E and forming their composite pv: 1 --> C^E as mu_E T(p) v: 1 --> C^N --> C^(C^E) --> C^E. The overflow was occurring in T(p): C^N --> C^(C^E); the particular measurement happened to depend only on finitely many of the N dimensions of v so naturally the programmer had organized p to map all the unused dimensions to 0 as a single element of the set C^E. T(p): C^N --> C^(C^E) maps v: C^N to u: C^(C^E) formed by initializing u to 0 and then for each i in N adding v_i to u_{p(i)}. Since p(i) = 0 for all but finitely many i, whenever v is divergent (not contradicted by square summability) coordinate 0 of u will overflow. So the overflow problem was happening even before mu kicked in (otherwise mu would have saved it). When the Haskell programmers came in on Wednesday to get started on programming and found only an overflowing machine, they looked askance at C^(C^E) and said "That's ridiculous, no wonder it doesn't work." So they jury-rigged the Haskell realization of monad in place of the Kleisli definition. This replaces the multiplication of the monad and its deployment in the Kleisli construction by Haskell's Bind operator typed as T(X) --> ((X --> T(Y)) --> T(Y)). In our machine this becomes C^N --> ((N --> C^E) --> C^E). Bind combines the separate actions of T(p) and mu_E into one by setting the coefficient v_e of output v: C^E to the sum of v_i*p_{ie} over all i in N. It thereby reverses the previous order of adding and multiplying, multiplying the coefficients of v_i by p_{ie} *before* adding them, harmlessly zeroing out the unused cofinite portion of v. In the Kleisli implementation the unused coefficients first accumulated at location 0 of C^(C^E) and were *then* multiplied by that location (i.e. by the complex number zero), but too late because the addition had already overflowed. Both formulations of monad realize matrix multiplication, but in the infinite-dimensional case the direct application of the monad multiplication via Kleisli would seem more problematic than Haskell's Bind. On Thursday the EEs reported excitedly that Mv was converging just fine in the situations that had caused the old definition to take an overflow exception. Friday morning found marketing in bedlam. "If that's God's monad," they asked the category theorists sarcastically, "how does God handle these exceptions?" "But God's monad works, I tell you," said the category theorists. "Look, you have an adjunction F -| G with unit eta and counit epsilon, you compose as (GF, G epsilon F, eta), and voila, a monoid object in Set^Set. And the Kleisli construction has never given trouble before." Voila indeed. At an emergency meeting called (voici) Friday afternoon in lieu of the regular TGIF the CEO impressed on us all the uncertainty and gravity of the situation (his background was in physics) while optimistically interpreting it as giving us the ultimate competitive advantage: a computer more reliable than the universe. Being even more risk averse than the VCs however and ever mindful of the truth-in-advertising laws, he has assigned me to escalate the question of whether we've really improved on the universe to the categories mailing list for their opinion. Is the Maharishi mathematicians' monad God's real McCoy or just misinformed mathematics? Why does it fail where the Haskell Bind operator succeeds? And does Bind always work? We've only tested it on a few cases so far. Signed/sealed/delivered: Vaughan Pratt From rrosebru@mta.ca Mon Oct 8 10:13:57 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Oct 2007 10:13:57 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IesOJ-0003xO-Tr for categories-list@mta.ca; Mon, 08 Oct 2007 10:11:24 -0300 Date: Sun, 07 Oct 2007 16:20:42 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: Categories list Subject: categories: Re: Help! Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 21 I would think the best topics would be those that can be described with a minimum of jargon. The problem with category theory is that it is so steeped in its own jargon as to make it quite an effort to strip it out. Here are some topics where I would expect that effort to be minimal, arranged in roughly increasing order of intricacy of definition. This should more than fill a one-hour lecture, especially if there are questions. 1. Thinking of each object T of a category C as both a type and a dual type, characterize a product AxB in C as an object consisting of all pairs of T-elements of A and B over all types T, and A+B as dually consisting of all pairs of T-functionals of A and B over all dual types T in ob(C). Section 6 needs pullbacks, they could be done either here or there, it's neither here nor there. 2. The category FinBip of finite bipointed sets as the theory of cubical sets. The models are arbitrary functors M: Bip --> Set. You could look at \Delta for simplicial sets as well or instead, I'm partial to Bip perhaps because in kindergarten we tended to work more with cubical than simplicial sets (Western Australian kindergartens had strong PTOs reflecting epic entanglements). You could then continue with FinSet^op as the algebraic theory of Boolean algebras, but that would entail giving up one of the other segments. 3. Enriched categories as generalized metric spaces. People who have a hard time with abstract objects mixed in with concrete homsets (I certainly did) will be relieved to know that making the homsets just as abstract as the objects turns the definition of category into a familiar object not normally considered part of the categorical basement. 4. Presheaves on J as colimits of diagrams in J. If you use Yoneda to hide the concept of colimit the idea becomes almost trivial. In the case J = 1, as C starts from 1 and grows towards Set^1 each new set X as a new object of C is installed along with an arbitrary choice of C(1,X). The composites at X are defined by first installing C(X,Y) for all existing Y in C, defining fx: 1 --> X --> Y for each x in X and new f in C(X,Y), and taking C(X,Y) to be maximal subject to Ax[fx=gx] ==> f=g. These composites then uniquely determine the remaining composites gf: X --> Y --> Z and fg: W --> X --> Y for W other than 1. The completion is complete when every new set is necessarily isomorphic to one already present. (Does this have anything to do with Yoneda structures? Trying to read about those I discovered I no longer talked Strine.) For J the ordinals 1 and 2 as respectively the primitive vertex and the primitive edge, namely the two reflexive graphs priming the pump for the rest, there are now two types of element, vertices and edges, with Ax interpreted as quantifying over all elements of both types; otherwise everything is as for J = 1. Point out that whereas all sets are free, the free graphs are just those with trivial incidences. If you do section 6 (triples for matrix multiplication), also point out at some point that whereas Set^1 is tripleable on Set (the identity), Set^J in general is tripleable only on Set^{|J|}, important when talking Czech. 5. Toposes, but *not* the way it is explained on You-tube, which is completely unmotivated and incomprehensible for anyone who hasn't already understood them. The Explanation section http://en.wikipedia.org/wiki/Topos_theory#Explanation in the Wikipedia article on elementary toposes touches on the two points that should be in any explanation of the concept, namely (i) "subobject" predates "topos," witness CTWM which defines it in second-order language, and (ii) monics m: X' --> X are in bijection with pullbacks of the element (hence monic) t: 1 --> \Omega along morphisms f: X --> \Omega, allowing one to speak of *the* characteristic function of a monic, thereby classifying the monics by their characteristic functions, a first-order notion (whence the "elementary" in "elementary topos") that is in full agreement with the second-order notion in (i) when applied to a topos. 6. Matrix multiplication in terms of the Kleisli construction for the triple for Vct_k. I just sent out a crib sheet for that which focused on a difficulty with non-finitary (square summable) linear combinations, but that difficulty is impossible to absorb in the available time, better to stick to the finitary operations where matrix multiplication is tripleable. You could mention the Haskell programming language and how they blended the second component of the triple and Kleisli into a single operator Bind: T(X) --> (X --> T(Y)) --> T(Y), which might get any programmers in the club interested in Haskell; also point out the possibility of replacing (X --> T(Y)) by T(Y*X) and its implications for matrix algebra including Hilbert space. Stick to finite X in T(X) = k^X to save the extra step of defining finitary k^X for infinite X (but if you do decide to do that step it should suffice to point out that 6 of the 16 binary Boolean operations as 2x2 truth tables have constant rows or columns or both and then generalize to infinity). In case You-tube ever has a video on triples you should probably mention any synonyms for "triple" so the students can find the video. Vaughan Michael Barr wrote: > What would you say to an undergraduate math club about categories? I have > been thinking about it, but I am not sure what to say. Talk about > cohomology, which is what motivated E-M? I don't think so. Talk about > dual spaces of finite-dimensional vector spaces? Maybe, but then what? > > Michael > > > From rrosebru@mta.ca Mon Oct 8 10:13:58 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Oct 2007 10:13:58 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IesKJ-0003g8-6a for categories-list@mta.ca; Mon, 08 Oct 2007 10:07:15 -0300 Date: Sun, 07 Oct 2007 17:25:37 +0100 From: Luke Ong MIME-Version: 1.0 To: categories@mta.ca Subject: categories: TCS 2008: 1st Call for Papers Content-Type: text/plain; charset=UTF-8; format=flowed Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 22 First Announcement and Call for Papers 5th IFIP International Conference on Theoretical Computer Science (TCS-2008) http://www.aicanet.it/wcc2008/TCS2008cfp1.pdf Held in conjunction with the 20th IFIP World Computer Congress September 7-10, 2008, Milano, Italy Conference Co-Chairs: Giorgio Ausiello, IT Giancarlo Mauri, IT Programme Co-Chairs: Track A: Juhani Karhum=C3=83=C2=A4ki, FI Track B: Luke Ong, GB Programme Committee: Track A: Algorithms, Complexity & Models of Computation Ricardo Baeza-Yates (Santiago), Marie-Pierre Beal (Paris),Harry Buhrman (Amsterdam), Xiaotie Deng (Hong Kong), Josep Diaz (Barcelona), Volker Diekert (Stuttgard), Manfred Droste (Leipzig), Ding-zhu Du (Dallas), Juraj Hromkovic (Zurich), Oscar Ibarra (Santa Barbara), Pino Italiano (Rome), Kazuo Iwama (Kyoto), Juhani Karhum=C3=83=C2=A4ki (Turku, chair), = Pekka Orponen (Helsinki), George Paun (Bucharest), Jiri Sgall (Prague), Alexander Shen (Moscow), Vijai Vazirani (Atlanta), Mikhail Volkov (Ekaterinburg) Track B: Logic, Semantics, Specification and Verification Rajeev Alur (Pennsylvania), Ulrich Berger (Swansea), Andreas Blass (Ann Arbor), Anuj Dawar (Cambridge), Mariangiola Dezani-Ciancaglini (Turin), Gilles Dowek (Paris), Peter Dybjer (Stockholm), Masami Hagiya (Tokyo), Martin Hofmann (Munich), Leonid Libkin (Edinburgh), Huimin Lin (Beijing), Stephan Merz (Nancy), Dale Miller (Paris), Eugenio Moggi (Genova), Anca Muscholl (Bordeaux), Luke Ong (Oxford, chair), Davide Sangiorgi (Bologna), Thomas Schwentick (Dortmund), Thomas Streicher (Darmstadt), P. S. Thiagarajan (Singapore), Wolfgang Thomas (Aachen) Submission will be in two stages: a short abstract due on 8 February and the 12-page paper due on 15 February 2008. The results of the paper must be unpublished and not submitted for publication elsewhere, including journals and the proceedings of other symposia or workshops. Authors will be notified of acceptance or rejection via e-mail by 31 March. Full versions of accepted papers (camera-ready) must be written in English, and will be due by 22 April 2008. One author of each accepted paper should present it at the conference. Scope and Topics TCS2008 will be composed of two distinct, but interrelated tracks: Track A on Algorithms, Complexity and Models of Computation, and Track B on Logic, Semantics, Specification and Verification. Suggested, but not exclusive, topics of interest include: Track A - Algorithms, Complexity and Models of Computation Analysis and design of algorithms; Automata and formal languages; Cellular automata and systems; Combinatorial, graph and optimization algorithms; Computational learning theory; Computational complexity; Computational geometry; Cryptography; Descriptive complexity; Evolutionary and genetic computing; Experimental algorithms; Mobile computing; Molecular computing and algorithmic aspects of bioinformatics; Network computing; Neural computing; Parallel and distributed algorithms; Probabilistic and randomized algorithms; Quantum computing; Structural information and communication complexity. Track B - Logic, Semantics, Specification and Verification Automata theory; automated deduction; constructive and non-standard logics in computer science; concurrency theory and foundations of distributed and mobile computing; database theory; finite model theory; formal aspects of program analysis, foundations of hybrid and real-time systems; lambda and combinatory calculi; logical aspects of computational complexity; modal and temporal logics; model checking and verification; probabilistic systems; logics and semantics of programs; foundations of security; term rewriting; specifications; type, proof and category theory in computer science. Paper submission Papers presenting original research in conference topics are being sought. The proceedings will be published by SSBM (Springer Science and Business Media). Submissions, as well as final versions, are limited to 12 pages, in the final SSBM format. The instructions for preparing the papers can be downloaded from http://www.springeronline.com/sgw/cda/frontpage/0,11855,4-40492-0-0- or ftp://ftp.springer.de/pub/tex/latex/ifip/ . Only electronic submissions will be accepted, via Track A: http://www.easychair.org/TCS2008-TrackA Track B: http://www.easychair.org/TCS2008-TrackB The submission deadline, length limitations and formatting instructions are firm: any submissions that deviate from these may be rejected without further considerations. IMPORTANT DATES: 8 February 2008: Abstract submission deadline 15 February 2008: 12-page paper submission deadline 31 March 2008: Notification of acceptance 7 April 2008: Copyright release submission deadline 22 April 2008: Camera-ready copy submission deadline Organized by IFIP Technical Committee 1 (Foundations of Computer Science) and IFIP WG 2.2 (Formal Descriptions of Programming Concepts) in association with SIGACT and EATCS From rrosebru@mta.ca Mon Oct 8 10:13:58 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Oct 2007 10:13:58 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IesIy-0003aU-KG for categories-list@mta.ca; Mon, 08 Oct 2007 10:05:52 -0300 Date: Sun, 7 Oct 2007 10:43:23 -0400 (EDT) From: Michael Barr To: Categories list Subject: categories: Re: Help! MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 23 I want to thank all who replied and I will take all your comments seriously. I would love to talk about Stone duality and such but I don't think many of our undergrads have ever heard of a topological space. They have heard of topology of course, but mostly they think it concerns things like toruses and Klein bottles. So they know nothing of the point set underpinnings of algebraic topology. My last term before retirement I taught a course called topology and spent exactly 6 lectures on point-set theory (taking a beeline to the Tychonoff theorem) before introducing pi_1 and covering spaces. The students were last year undergrads and a couple of grad students. Do they know what a boolean algebra is? Probably some do, some don't. Groups and abelian groups they will know about, probably modules, etc. Vector space duality is a familiar example, for finite dimension at least. Hi-tech whiteboards and even video-taping are out. I don't think we have any of the former and the one case that I know of a lecture that was video-taped (a fascinating lecture by Conway in the early '70s in which he showed how the game of Life allowed the simulation of self-reproducing Turing-power automata) seems to have disappeared without a trace. I will probably use a blackboard (or greenboard) and chalk, my favorite medium. One suggestion that does appeal is to start with universal mapping properties to explain products and sums. One thing that always struck me was Bill Lawvere's observation that the dual of the usual definition of function as a subset of a product s.t.... namely as a quotient of a sum s.t.... actually corresponds closely to the usual picture we draw when we introduce functions for the first time. I guess I could do worse than build the whole lecture around universal mapping properties. I could mention the somewhat unmotivated definition of (infinite) product of topological spaces as a perfect example of the universal viewpoint. Especially as topologists had come up with that definition independent of category theory. Michael From rrosebru@mta.ca Mon Oct 8 10:13:58 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Oct 2007 10:13:58 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IesGs-0003Q6-6n for categories-list@mta.ca; Mon, 08 Oct 2007 10:03:42 -0300 Date: Fri, 05 Oct 2007 23:42:46 +0100 To: categories@mta.ca Subject: categories: Announcement: Ackermann Award 2007 MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit From: A.Beckmann@swansea.ac.uk (Arnold Beckmann) Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 24 [Apologies for multiple copies] ======================================================================== EACSL - The European Association of Computer Science Logic September, 2007 2007 Ackermann Award of the EACSL --------------------------------- EACSL Homepage: http://www.dimi.uniud.it/~eacsl/ Ackermann Award Homepage: http://www.cs.technion.ac.il/eacsl/ackermann/ The Jury of the Ackermann Award has decided to give the 2007 Ackermann Awards to Dietmar Berwanger RWTH Aachen (Advisor: Erich Graedel) http://mtc.epfl.ch/~dwb/ Thesis: Games and Logical Expressiveness Stephane Lengrand Universite de Paris VII and University of St. Andrews (Advisors: Delia Kesner and Roy Dyckhoff) http://www.pps.jussieu.fr/~lengrand/ Thesis: Normalization and Equivalence in Proof Theory and Type Theory Ting Zhang Stanford University (Advisor: Zohar Manna) http://theory.stanford.edu/~tingz/ Thesis: Arithmetic Integration of Decision Procedures I would like to congratulate the recipients and their supervisors for their excellent theses. Previous Ackermann Award recipients were: 2005: Mikolaj Bojanczyk, Konstantin Korovin, Nathan Segerlind; 2006: Stefan Milius and Balder ten Cate; The Jury consisted of S. Abramsky, J. van Benthem, B. Courcelle, M. Grohe, M. Hyland, J. Makowsky, D. Niwinski, A. Razborov. The Award Ceremony took place during the CSL'07 Conference. http://www.inf.u-szeged.hu/~csl06/ A detailed report is published in the CSL'07 Proceedings. I would like to thank all the Jury members for their work. J.A. Makowsky President of EACSL and chairman of the Jury http://www.cs.technion.ac.il/~janos EACSL WEB-pages: http://www.dimi.uniud.it/~eacsl/ http://www.cs.technion.ac.il/eacsl/ http://www.cs.technion.ac.il/eacsl/ackermann/ ================================================================ From rrosebru@mta.ca Mon Oct 8 10:13:58 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Oct 2007 10:13:58 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IesGD-0003O1-Bb for categories-list@mta.ca; Mon, 08 Oct 2007 10:03:01 -0300 To: categories@mta.ca Subject: categories: MPC 2008: FIRST CALL FOR PAPERS Reply-To: Ch.Paulin@lri.fr From: Christine Paulin Date: Wed, 03 Oct 2007 10:34:30 +0200 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 25 FIRST CALL FOR PAPERS 9th International Conference on Mathematics of Program Construction (MPC'08= ) Marseille (Luminy), France, July 15-18th 2008 http://mpc08.lri.fr BACKGROUND The biennial MPC conferences aim to promote the development of mathematical principles and techniques that are demonstrably practical and effective in the process of constructing computer programs. Topics of interest range from algorithmics to support for program construction in programming languages and systems. The previous conferences were held in Twente, The Netherlands (1989), Oxford, UK (1992), Kloster Irsee, Germany (1995), Marstrand, Sweden (1998), Ponte de Lima, Portugal (2000), Dagstuhl, Germany (2002), Stirling, UK (2004, colocated with AMAST '04) and Kuressaare, Estonia (2006, colocated with AMAST '06). The 2008 conference will be held in Marseille, France at the International Center for Mathematical Meetings (http://http://www.cirm.univ-mrs.fr/web.ang). INVITED SPEAKERS To be announced. IMPORTANT DATES * Submission of abstracts: 14 January 2008 * Submission of full papers: 21 January 2008 * Notification of authors: 10 March 2008 * Camera-ready version: 10 April 2008 TOPICS Papers are solicited on mathematical methods and tools put to use in program construction. Topics of interest range from algorithmics to support for program construction in programming languages and systems. Some typical areas are type systems, program analysis and transformation, programming-language semantics, program logics. Theoretical contributions are welcome provided their relevance for program construction is clear. Reports on applications are welcome provided their mathematical basis is evident. SUBMISSION Submission is in two stages. Abstracts (plain text) must be submitted by 14 January 2008. Full papers (pdf) adhering to the llncs style must be submitted by 21 January 2008. There is no official page limit, but authors should strive for brevity. The web-based submission system will open in early December 2007. Papers must report previously unpublished work and not be submitted concurrently to another conference with refereed proceedings. Accepted papers must be presented at the conference by one of the authors. The proceedings of MPC'08 will be published in the Lecture Notes in Computer Science series of Springer-Verlag. After the conference, the authors of the best papers will be invited to submit revised versions to a special issue of the Science of Computer Programming journal of Elsevier. PROGRAMME COMMITTEE Christine Paulin-Mohring INRIA-Universit=E9 Paris-Sud, France (chair) Philippe Audebaud=09Ecole Normale Sup=E9rieure Lyon, France (co-chair) Ralph-Johan Back=09Abo Akademi University,=09Finland Eerke Boiten=09=09University of Kent, UK Venanzio Capretta=09University of Nijmegen, Netherlands Sharon Curtis=09=09Oxford Brookes University, UK Jules Desharnais=09Universit=E9 Laval, Qu=E9bec, Canada Peter Dybjer=09=09Chalmers University of Technology, Sweden Jeremy Gibbons =09=09University of Oxford, UK Lindsay Groves=09=09Victoria University of Wellington, New Zealand Ian Hayes=09=09University of Queensland, Australia Eric Hehner=09=09University of Toronto, Canada Johan Jeuring =09=09Utrecht University, Netherlands Dexter Kozen =09=09Cornell University, USA Christian Lengauer=09Universit=E4t Passau, Germany Lambert Meertens=09University of Utrecht, Netherlands Bernhard M=F6ller =09Universit=E4t Augsburg, Germany Carroll Morgan=09=09University of New South Wales, Australia Shin-Cheng Mu=09=09Academia Sinica, Taiwan Jose Nuno Oliveira =09Universidade do Minho, Portugal Tim Sheard=09=09Portland State University, USA Tarmo Uustalu =09=09Institute of Cybernetics Tallin, Estonia VENUE The conference will be held in Marseille, the second largest city in France next to Paris. Its port is the most important in France, and opens the city to the world through the Mediterranean Sea. MPC'08 will be hosted by the International Center for Mathematical Meetings. The center is located inside the Campus of Luminy Faculty. It is close to the "Calanques", an astounding wild coastline composed of creeks stretching from Marseille to Cassis. LOCAL ORGANIZERS MPC 2008 is organized with the support of INRIA. The local organizers are Philippe Audebaud and Christine Paulin-Mohring. Enquiries regarding the programme (submission etc.) should be addressed to mpc08(at)lri.fr From rrosebru@mta.ca Mon Oct 8 10:13:58 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Oct 2007 10:13:58 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IesNG-0003sr-Vc for categories-list@mta.ca; Mon, 08 Oct 2007 10:10:19 -0300 Date: Sun, 7 Oct 2007 22:49:22 +0100 (BST) From: "Prof. Peter Johnstone" To: Categories Subject: categories: Re: "Historical terminology" MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 26 On Sun, 7 Oct 2007, Jean Benabou wrote: > (ii) Your "guess" about cartesian is not correct. Neither in Tohoku, > nor in much later papers of his or any of his students, and also by > me, was cartesian used in the sense of category with finite limits. > If Grothendieck had used this > name, which he has not, my "guess" is that he would have called > cartesian categories with pull backs , because he and his students > used the name "cartesian square" for square which is a pull back. > Moreover this is special case of his notion of cartesian map in > a fibration. > I first encountered `cartesian' as a synonym for `having finite limits' in Peter Freyd's unpublished `pamphlet' "On canonizing category theory; or, on functorializing model theory" written in about 1975 (I may have got the title wrong, since I no longer possess a copy). However, that paper made it clear that the word was already in use as a synonym for "having finite products"; in it, Peter argued that Descartes should be given credit for having invented equalizers as well as cartesian products. I suspect that its use to mean `having finite products' was a conscious back-formation from `cartesian closed', which undoubtedly dates from Eilenberg--Kelly 1965; but I don't know who first used it in this sense. > (iii) I agree with you on the idea that the "natural" definition of > locally cartesian closed category should not imply the existence > of a terminal object. If I asked the question, it is because in > Johnstone's "Elephant" he does assume a terminal object. Has such an > assumption become, now, commonly accepted in the definition ? > I did that because it seemed the appropriate convention to adopt in the context of topos theory. I wasn't trying to dictate to the rest of the world what the convention should be. On the other hand, there seem to be remarkably few `naturally occurring' examples of locally cartesian closed categories which lack terminal objects: the category of spaces (or locales) and local homeomorphisms is almost the only one I can think of. Peter Johnstone From rrosebru@mta.ca Mon Oct 8 10:13:58 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Oct 2007 10:13:58 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IesLW-0003kQ-Av for categories-list@mta.ca; Mon, 08 Oct 2007 10:08:30 -0300 Date: Sun, 7 Oct 2007 10:11:03 -0700 From: Toby Bartels To: Categories list Subject: categories: Re: Help! MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 27 Michael Barr wrote: >What would you say to an undergraduate math club about categories? I have >been thinking about it, but I am not sure what to say. Talk about >cohomology, which is what motivated E-M? I don't think so. Talk about >dual spaces of finite-dimensional vector spaces? Maybe, but then what? When I was a graduate student (recently), I gave a talk on category theory to other (mostly new) grad students (as part of a series where advanced students discussed their work). I began with my definition of category theory for nonmathematicians ("a general theory of how mathematical structures can fit together"), then gave some basic definitions and an example (duality in finite-dimensional vector spaces). Then I asked the audience a very open-ended question: Tell me what's your favourite branch of mathematics, and I'll tell you what category theory has to say about it (to justify the generality in my beginning statement). What attracted me first to category theory, and what I think remains impressive about it, is that you can you can really make good on this challenge. (It helps to know ahead of time what answers are likely; fortunately there were no pure number theorists at my school.) --Toby Bartels From rrosebru@mta.ca Mon Oct 8 10:13:58 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Oct 2007 10:13:58 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IesPG-00043A-A4 for categories-list@mta.ca; Mon, 08 Oct 2007 10:12:22 -0300 Date: Sun, 07 Oct 2007 16:34:45 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories list Subject: categories: Re: Ideal Theory 101 [was: is 0 prime?] Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 28 wlawvere@buffalo.edu wrote: > The awesome nature of Sup cannot be the reason why > the Kummer functor exists, since it is merely used for=20 > recording the result. The functor is "caused" rather=20 > by an internal feature of the domain category C of=20 > commutative rings: The category of quotient objects > of any given R has a binary operation * that is neither > sup nor inf even though in principle it can be=20 > expressed as a combination of limits and colimits. Hear, hear. As a case in point the category Div that I described, namely the division category replacing the division lattice, has the number lcm(m,n) (least common multiple) as pushout over the categorical product gcd(m,n). This pushout is not the categorical sum of m and n, which is instead the number mn. (It is hell dealing with sum and product switching around like that down below. In the upper half of Div, namely FinSet, sum is m+n and product is mn as it is in heaven.) > We can call it R/ab=3DR/a *R/b but how does the=20 > operation * specialize to C concretely ? I was wondering the same thing. I bet something good would come out of a meeting between category theorists and ring theorists on the topic of finding the right abstractions here---presumably a lot of the groundwork is already in place, much as it was for UACT in 1993, although as I recall the algebraists didn't seem in the mood at the time. Vaughan From rrosebru@mta.ca Mon Oct 8 10:14:11 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Oct 2007 10:14:11 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IesQR-00047n-EG for categories-list@mta.ca; Mon, 08 Oct 2007 10:13:35 -0300 Date: Sun, 07 Oct 2007 17:12:36 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: Categories Subject: categories: Re: "Historical terminology" Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 29 JeanBenabou wrote: > (i) Your "guess" about cartesian closed categories is most certainly > correct. I knew that Eilenberg/Kelly had explicitly used this name > in their La Jolla paper, and it is probably the first instance, > because "closed", in this sense, was first introduced in that paper, > as far as I know.. What most impressed my students and me two decades ago, when we were applying the concepts of EK65 to modeling concurrency, was their attempt to define "closed" as a self-contained notion independently of any tensor product as its left adjoint (or so it seemed to us). This defeated us. Has a clearer story of that attempt, or any related story, emerged in the meantime? > (iii) I agree with you on the idea that the "natural" definition of > locally cartesian closed category should not imply the existence > of a terminal object. If I asked the question, it is because in > Johnstone's "Elephant" he does assume a terminal object. Has such an > assumption become, now, commonly accepted in the definition ? Hopefully not. If affine geometry has no origin, why should locally cartesian closed categories have a global reference point? (What would Andy Pitts have decided there, and for that matter the orientation of profunctors in B2.7, which seems backwards from say Borceux?) Vaughan From rrosebru@mta.ca Mon Oct 8 10:15:46 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Oct 2007 10:15:46 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IesRx-0004Ii-03 for categories-list@mta.ca; Mon, 08 Oct 2007 10:15:09 -0300 Date: Mon, 8 Oct 2007 07:10:36 +0200 (CEST) From: Mikael Vejdemo Johansson To: Categories list Subject: categories: Re: Help! MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 30 On Fri, 5 Oct 2007, Michael Barr wrote: > What would you say to an undergraduate math club about categories? I have > been thinking about it, but I am not sure what to say. Talk about > cohomology, which is what motivated E-M? I don't think so. Talk about > dual spaces of finite-dimensional vector spaces? Maybe, but then what? > How about talking about simultaneously existing results in several categories? The Noetherian isomorphism theorems, while not necessarily the easiest to nail down exactly when they hold, have always been a strong motivator at the back of my head for why one might want to look at algebraic entities codifying things like "All Xs and maps between them". -- Mikael Vejdemo Johansson | To see the world in a grain of sand mik@math.su.se | And heaven in a wild flower | To hold infinity in the palm of your hand | And eternity for an hour From rrosebru@mta.ca Mon Oct 8 10:17:50 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Oct 2007 10:17:50 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IesU4-0004Uc-B5 for categories-list@mta.ca; Mon, 08 Oct 2007 10:17:20 -0300 Date: Mon, 8 Oct 2007 16:38:24 +1000 From: "Micah Blake McCurdy" To: categories@mta.ca Subject: categories: Talking to Undergraduates about Category Theory MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 31 Hallo! I have over the last several years repeatedly given to delegates of the Canadian Undergraduate Mathematics Conference a talk about Category Theory, all of which were very well received. I should mention at the outset that I had the (debatable) advantage of _being_ an undergraduate for all three talks. Some elements which went over especially well: 1) Historical considerations, namely, the role of category theory in the development of algebraic topology. The use of category theory as a language for making rigorous certain intuitions, as well as facilitating calculations. This point of view resonates very well with undergraduates, to whom _all_ of mathematics is a more or less hazy mass of intuitions and proofs; who long for clarity and order. 2) Freeing constructions from set by diagrammatic descriptions. For instance, defining the notion of a group object in an arbitrary category C, and then noting that such gadgets are already studied for various C. This appeals for two reasons: it gives an elegant explanation for _why_ similar-seeming things are similar, and, more importantly, it _suggests new questions_, namely, for a new category of study, "what are the internal wombats in this category" for various choices of wombat. Especially for older undergraduates, who are thinking to themselves "Subject X is really very fascinating, but what will I ever do with it?", this is a very appealing notion. 3) Diagrammatic methods in proofs. The device of commuting diagrams to form and illustrate proofs is generally both novel and wonderful to undergraduates. This has many sub-parts, among them: i) One augments a symbolic intuition with a geometric intuition. Thus, proving that a large diagram commutes becomes a sort of tangram puzzle. ii) Proofs become both easier to construct and, more importantly, easier to communicate. This is especially near to the hearts of undergrads who have difficulty constructing proofs, more difficulty understanding the proofs of others, and yet more in having their proofs understood by others. If you were so inclined, you might well introduce string diagrams. The third point, considered strictly, is not really a part of category theory, but I think it is cut from the same cloth. On a perfectly peripheral note, I often place two bottles of (preferably obscure) beer on the desk in front of me before I begin speaking; promising one to the best question after the talk and the other to the best heckling during the talk. I strongly encourage heckling, and I doubt that undergraduates enjoy this any more than other mathematicians. If all goes wrong, you can drink the beer yourself. In any event, good luck. Micah From rrosebru@mta.ca Mon Oct 8 10:18:41 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 08 Oct 2007 10:18:41 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IesUn-0004Xz-Dz for categories-list@mta.ca; Mon, 08 Oct 2007 10:18:05 -0300 Date: Mon, 08 Oct 2007 10:34:35 +0100 From: John Power To: categories@mta.ca Subject: categories: New contact details MIME-Version: 1.0 Content-Type: text/plain;charset=ISO-8859-1;format="flowed" Content-Disposition: inline Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 32 Dear All, My new contact details are as follows: Email: A.J.Power@bath.ac.uk. (You should now use that rather than any other email address you might have for me.) Address: Department of Computer Science University of Bath Claverton Down, Bath BA2 7AY United Kingdom Tel: +44 1225 384439 All the best, John. From rrosebru@mta.ca Tue Oct 9 00:04:12 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Oct 2007 00:04:12 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1If5I4-0007ja-GV for categories-list@mta.ca; Mon, 08 Oct 2007 23:57:48 -0300 Date: Mon, 08 Oct 2007 11:18:42 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: Categories Subject: categories: Re: "Historical terminology" Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 33 Vaughan Pratt wrote: > JeanBenabou wrote: >> (i) Your "guess" about cartesian closed categories is most certainly >> correct. I knew that Eilenberg/Kelly had explicitly used this name >> in their La Jolla paper, and it is probably the first instance, >> because "closed", in this sense, was first introduced in that paper, >> as far as I know.. > > What most impressed my students and me two decades ago, when we were > applying the concepts of EK65 to modeling concurrency, was their attempt > to define "closed" as a self-contained notion independently of any > tensor product as its left adjoint (or so it seemed to us). This > defeated us. Has a clearer story of that attempt, or any related story, > emerged in the meantime? Meanwhile the following examples occurred to me: 1. Implicational logic without conjunction. 2. The type structure of the pure lambda calculus without products. 3. The subcategory of FinSet consisting of the prime powers. (With regard to 3, Mike Barr mentioned to me that (Eilenberg and?) Kelly had come up with the category "-6" meaning the category of all sets save those with six elements, but this seems less natural than the prime powers, important in ideal theory as we saw in the recent discussion about the division lattice.) The free closed category would be a good example if it had ever been sighted in nature? Has it? (Just because we see initial ring every day in the wild doesn't mean that all free objects arise in nature.) Vaughan From rrosebru@mta.ca Tue Oct 9 00:04:12 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Oct 2007 00:04:12 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1If5Fk-0007bC-Ey for categories-list@mta.ca; Mon, 08 Oct 2007 23:55:24 -0300 Mime-Version: 1.0 (Apple Message framework v624) Content-Type: text/plain; charset=US-ASCII; format=flowed Content-Transfer-Encoding: 7bit From: Paul Taylor Subject: categories: locally cartesian closed categories Date: Mon, 8 Oct 2007 16:28:37 +0100 To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 34 I agree with Jean Benabou, Fred Linton and Vaughan Pratt that the definition of a locally cartesian closed category should NOT require a terminal object. I expressed this view in a footnote on page 499 of "Practical Foundations of Mathematics", with a justification similar to Vaughan's. The simplest formulation is that an LCCC is a category every slice of which is a CCC. In particular, every slice has binary products, which are pullbacks in the whole category. Objects of an LCCC and the slices that they define correspond to objects of a base category and the fibres over them in a fibred or indexed formulation of logic, and to contexts in a syntactical one. Contexts are collections of hypotheses. The terminal object or empty context is the one with no hypotheses at all. However, as the 18th century logician Johann Lambert remarked, ``no two concepts are so completely dissimilar that they do not have a common part''. If "naturally occurring" LCCCs usually have terminal objects, I suggest that that may be because they are already slices of some more general picture. For more or less the same reason, we never actually concatenate two contexts, but build them up one hypothesis at a time. So I would say that, whilst an LCCC has pullbacks, it need not have binary products. (My footnote refers to "other authors" who said that LCCCs should have binary products; I think I may have had Thomas Streicher in mind, but I don't recall what he may have said or in what paper.) I confess that I'm a bit surprised to find that the consensus agrees with me, so to set matters straight I should also point out that my argument applies equally to elementary toposes and other familiar structures of categorical logic. ---- While we're playing around with the structures of categorical logic, let me try another related question. Any topos is a CCC with an internal Heyting algebra. [WARNING TO STUDENTS: whilst this statement is true, it's NOT (equivalent to) the correct definition.] I am sorry to say that I have seen papers emanating from respectable universities in which the authors have appeared to believe that this is the definition. (One of the papers that I have in mind cites many eminent categorists, who may perhaps have an opinion about having their names appear alongside a lot of complete nonsense.) But I wonder whether anyone has taken this idea seriously, and investigated how much logic such a category would admit? The version of this question that particularly interests me is this: Suppose that the category has all FINITE LIMITS (terminal object, finite products and equalisers) and POWERS Sigma^X of an internal DISTRIBUTIVE LATTICE (Sigma, top, bot, meet, join). Maybe there is also a natural numbers object N and joins Sigma^N->Sigma with the Frobenius law. (I would also like this to obey the monadic and Phoa principles of ASD, but I'm not going to spell them out here.) Maps X->Sigma give rise to a "geometric logic" of "open" subspaces. Then the order relation between maps X->Sigma^Y leads to a richer logic of "general" subspaces, with => and forall_Y. A logical formula of the more general form consists of geometric sub-formulae joined together with => and forall, to which we might add the other first order connectives as "syntactic sugar", defined in the usual classical way. If a geometric sub-formula is immediately enclosed in forall_K or exists_N, where K happens to be compact or N overt, then this a priori more general quantifier may be considered to be part of the geometric sub-formula. Does this idea ring any bells? Paul Taylor From rrosebru@mta.ca Tue Oct 9 00:04:12 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Oct 2007 00:04:12 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1If5GP-0007dN-MU for categories-list@mta.ca; Mon, 08 Oct 2007 23:56:05 -0300 Date: Mon, 08 Oct 2007 08:43:15 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: Categories list Subject: categories: Re: Help! Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 35 > (It helps to know ahead of time what answers are likely; > fortunately there were no pure number theorists at my school.) At the risk of sounding like a cracked record, how about the division category in lieu of the division lattice, namely the coproduct completion of the set P of primes as a discrete category? For a longer story use P* instead of P, P with a final object adjoined. Motivate the division category by pointing out that only the square-free positive integers can be recovered as sups of primes in the division lattice. Vaughan From rrosebru@mta.ca Tue Oct 9 00:04:13 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Oct 2007 00:04:13 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1If5Ia-0007le-1H for categories-list@mta.ca; Mon, 08 Oct 2007 23:58:20 -0300 Date: Mon, 8 Oct 2007 11:48:17 -0700 From: Toby Bartels To: Categories list Subject: categories: Re: Help! MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 36 Vaughan Pratt wrote at last part: >In case You-tube >ever has a video on triples you should probably mention any synonyms for >"triple" so the students can find the video. YouTube has a series of 5 video on triples under the name "monads": < http://www.youtube.com/results?search_query=monads&search=Search >, among others by the Catsters < http://www.youtube.com/user/TheCatsters >. --Toby From rrosebru@mta.ca Tue Oct 9 00:04:13 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Oct 2007 00:04:13 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1If5J2-0007mm-A4 for categories-list@mta.ca; Mon, 08 Oct 2007 23:58:48 -0300 Date: Mon, 08 Oct 2007 13:18:07 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories list Subject: categories: What is the right abstract definition of "connected"? Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 37 I'd like to say that "connected" is defined on objects of any category C having an object 1+1 (coproduct of two final objects). X is connected just when C(X,1+1) <= 2. If this definition appears in print somewhere I can just cite it. If not is there a better or more standard generally applicable definition I can use? If C(X,1+1) = 2 is citable but not <= 2, have the proponents of =2 taken into account that no Boolean algebra is connected according to the =2 definition? This is because 1+1 ~ 1 in Bool, CABA, DLat, StoneDLat, etc. (dual to 0x0 ~ 0 in Set, Pos, etc.), forcing C(X,1+1) = 1. Boolean algebras and distributive lattices fail the =2 test not because they are disconnected in any natural sense but rather because they are hyperconnected. It seems unreasonable to say that hyperconnected objects are not connected. There is also the question of the object of connected components of an object. In Set and Grph, if X has k connected components then C(X,1+1) = 2^k for all X, a set (C being ordinary, i.e. enriched in Set). In Stone (Stone spaces) however this only holds for finite X, with k = X. For infinite X Stone(X,1+1) is the set of clopen sets of X, which can be countably infinite and hence not 2^k for any k. If we read 2^k as Stone(k,2), taking k = X and 2 the Sierpinski space this doesn't help. However Stone(k,1+1) is ideal: instead of treating the object of connected components of a Stone space k = X as a set we can treat them as a Boolean algebra, namely that of the clopen sets of X. These examples are worth bearing in mind when considering the appropriate general definition of number of connected components of an object, and whether even to treat it as a number (cardinal) or a more general object. Connectedness seems somehow more basic than finiteness because we can easily draw examples of connected and disconnected objects, whereas it requires a vivid imagination to see the boundary between finite and infinite objects one might try to draw on paper. This motivates making connectedness prior to finiteness. Another familiar and easily visualized notion with small examples is that of path. Define a *path* to be a connected directed graph having one vertex each of degree (0,1) and (1,0), and all others (1,1). (The degree (m,n) specifies the in-degree as m and the out-degree as n.) We can then define a finite set to be one in bijection with the set of vertices of some path. This seems more natural than defining it to be one such that every injection on itself is a surjection, because there are a lot of injections to worry about and how do you convince yourself that surjective injections don't kick in until omega? Those who are already wedded to some other definition of finite will want to check that this path-based definition draws the boundary in the same place as theirs. For what definitions of "finite" can this not be shown? And are any of them more palatable than the path-based definition? Vaughan From rrosebru@mta.ca Tue Oct 9 00:04:13 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Oct 2007 00:04:13 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1If5FJ-0007XO-Lb for categories-list@mta.ca; Mon, 08 Oct 2007 23:54:58 -0300 Date: Mon, 8 Oct 2007 16:34:39 +0200 (MEST) From: Patrik Eklund To: Categories list Subject: categories: Re: Help! MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 38 On Sun, 7 Oct 2007, Michael Barr wrote: > Hi-tech whiteboards and even video-taping are out. I don't think we have > any of the former and the one case that I know of a lecture that was > video-taped (a fascinating lecture by Conway in the early '70s in which he > showed how the game of Life allowed the simulation of self-reproducing > Turing-power automata) seems to have disappeared without a trace. I will > probably use a blackboard (or greenboard) and chalk, my favorite medium. Hi Michael, Video-taping certainly is out, if it ever was in. Taping is non-interactive and just silly. Your attitude towards "Hi-tech whiteboards" sounds too hi-tech as my point indeed was to say what you say about your favourite medium. That is still also my favourite medium, but I accept to write or meet virtually in particular if my audience is a flight distance away. Something is lost when you go virtual, but you also win some. Do you resist virtual whiteboards per se, or would you be interested in trying out a session? Installation is less than 15 minutes, and once we are online, we could spend another 15 discussing idempotent functors extendable to monads where E-M and Kleisli coincide. The mouse is your chalk and your board colour is white. I've used it so much already over the last years so I cannot work without it anymore. I can supervise a student from my home or a hotel room in Tokyo, and nobody knows or even cares who's where. Cheers, Patrik PS And for those who didn't see my mail to Michael, here it is, and apologies to those who view this purely as spam: Date: Sun, 7 Oct 2007 07:19:34 +0200 (MEST) From: Patrik Eklund To: Michael Barr Cc: Patrik Eklund Subject: Re: categories: Help! Dear Michael, No comment (at this point) on content, but let me refer to a previous mail I sent out on the subject and related to execution. My idea was to suggest a setup of virtual classrooms so that students and teacher indeed all over the world can attend a class. Of course, students and teacher, and in the end content, must be carefully selected. The reason for my suggestion is that the number of students at many sites is usually bery low for these courses and we should join forces. My suggestion is to use "sound-video-whiteboard" techniques as provided e.g. by Adobe and Marratech. I use the latter. "Sound-video" is nothing but Skype, but adding whiteboards, that can be saved and worked with also offline, you have very good possibilities. The whiteboard mainly accepts non-formatted text, drawings and images. You can read doc and ppt file which are "pasted" as bitmaps on the whiteboard. They include desktop sharing if that would be required. Mathematical text I add through LaTeX, compiling, converting to pdf, and using the snapshot tool to paste bitmapped formulas on the whiteboard. Once you get used to it you are actually not (much) slower on the virtual whiteboard as compared to a real whiteboard. Virtual advantages are e.g. - several whiteboards and easy to switch between them - more than one can jointly add to whitebooard content - can save and open (as mentioned) - can prepare whiteboards offline (as mentioned) If this is inline with your thoughts and you would like to try out Marratech, let me know. Best, Patrik From rrosebru@mta.ca Tue Oct 9 00:04:13 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Oct 2007 00:04:13 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1If5HY-0007hL-63 for categories-list@mta.ca; Mon, 08 Oct 2007 23:57:16 -0300 Content-Disposition: inline Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset="us-ascii" From: To: Categories Subject: categories: Re: Historical terminology Date: Mon, 08 Oct 2007 14:02:35 -0400 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 39 Long before 1975, indeed before 1965, the term "cartesian product" was in use for topological spaces and for sets. At the latter date, the special case of monoidal closed structure in which the product is the CATEGORICAL one was dubbed cartesian. I do not recall whether it was Max or I who suggested it. (I was periferally involved in=20 the EK discussions because of the posetal case and my observations that logic is adjointness; the passing misnomer "Browerian " instead of "Heyting" was due to my reading of some papers by Tarski that used an odd convention). If I was responsible I regret it, since historical considerations suggest that Galileo contributed=20 more than Descartes toward crystallizing the catergorical product; especially the universality is strongly suggested by pairs of paths in a way that it to this day never was by pairs of mere points. Concerning the French usage they of course=20 knew that those interesting squares are just=20 products over a base, and I presume that the general case of fibrations was seen as a=20 generalization of the case C^2->C (induced=20 often as in algebraic geometry by some 2-functor from the latter), and hence generalized also the use of the terminology. Bill On Sun Oct 7 17:49 , "Prof. Peter Johnstone" sent: >On Sun, 7 Oct 2007, Jean Benabou wrote: > >> (ii) Your "guess" about cartesian is not correct. Neither in Tohoku, >> nor in much later papers of his or any of his students, and also by >> me, was cartesian used in the sense of category with finite limits. >> If Grothendieck had used this >> name, which he has not, my "guess" is that he would have called >> cartesian categories with pull backs , because he and his students >> used the name "cartesian square" for square which is a pull back. >> Moreover this is special case of his notion of cartesian map in >> a fibration. >> >I first encountered `cartesian' as a synonym for `having finite limits' >in Peter Freyd's unpublished `pamphlet' "On canonizing category theory; >or, on functorializing model theory" written in about 1975 (I may have >got the title wrong, since I no longer possess a copy). However, that >paper made it clear that the word was already in use as a synonym for >"having finite products"; in it, Peter argued that Descartes should be >given credit for having invented equalizers as well as cartesian products. >I suspect that its use to mean `having finite products' was a conscious >back-formation from `cartesian closed', which undoubtedly dates from >Eilenberg--Kelly 1965; but I don't know who first used it in this sense. > >> (iii) I agree with you on the idea that the "natural" definition of >> locally cartesian closed category should not imply the existence >> of a terminal object. If I asked the question, it is because in >> Johnstone's "Elephant" he does assume a terminal object. Has such an >> assumption become, now, commonly accepted in the definition ? >> >I did that because it seemed the appropriate convention to adopt in the >context of topos theory. I wasn't trying to dictate to the rest of the >world what the convention should be. On the other hand, there seem to >be remarkably few `naturally occurring' examples of locally cartesian >closed categories which lack terminal objects: the category of spaces >(or locales) and local homeomorphisms is almost the only one I can >think of. > >Peter Johnstone > > > > From rrosebru@mta.ca Tue Oct 9 22:00:10 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Oct 2007 22:00:10 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IfPpA-0001FN-1c for categories-list@mta.ca; Tue, 09 Oct 2007 21:53:20 -0300 Subject: categories: Re: locally cartesian closed categories From: Eduardo Dubuc Date: Tue, 9 Oct 2007 11:18:18 -0300 (ART) To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 40 hi, yet another point in favor that terminal object and products should not be mandatory in locally cartesian closed categories: terminal (or products) implies connection, fiber products don't. compare with the notion of cofilter category (axiom similar to existence of products), is connected, while pseudofiltered (axiom similar to existence of fiber products), is not connected. this is essentially the difference between filterness and cofilterness, with all what it means same thing, fiber products and not products are in the essence of the notion of locally cartesian closedness ps: congratulations to Bob R., I fully agree with all the good things that were said recently about his handling of this list (not an easy job !). eduardo dubuc From rrosebru@mta.ca Tue Oct 9 22:00:10 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Oct 2007 22:00:10 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IfPsJ-0001aN-5l for categories-list@mta.ca; Tue, 09 Oct 2007 21:56:35 -0300 Subject: categories: "role" vs. "r\^ole" To: categories@mta.ca (Categories List) Date: Tue, 9 Oct 2007 18:04:11 -0300 (ADT) MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit From: selinger@mathstat.dal.ca (Peter Selinger) Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 41 Hi everybody, here is a frivolous question only tangentially related to category theory. Does anyone know why it is common, in papers on logic, semantics, and category theory, to spell the word "role" the French way, i.e., with a circumflex accent? I am taking about the idiom "to play a role", as in, "in this definition, x and y play symmetric r\^oles". Sometimes it is also used as in "the r\^ole of x is ...". As far as I can tell, the accented spelling is a strange ideosyncrasy, given that the word "role", without the accent, is a perfectly acceptable, and very common, English word. Here are some examples I collected a few years ago: How big a role did politics play? -Los Angeles Times, March 27, 2002 Huge bomb could play role in Iraq. -The Guardian, March 13, 2003 Australia intends to play a role in [...] Iraq, The Australian, 4/15/2003 A movie in which Nicole Kidman could play the lead role -Business Times, 1/16/3 Genetics play a big role in your health. -Citizen-Times.com, April 11, 2003 Linux prepares to play broader role in embedded systems. -EETimes, 6/11/2001 The UN would play a central role in running the country. -Guardian, 4/10/2003 His role is to lead the paddlers through the race -Waterfront News, Oct 2007 How oil plays a role in an invasion of Iraq. -YellowTimes.org, Jan 22, 2003 I realize that Merriam Webster's Dictionary allows "r\^ole" as an alternate spelling (the Oxford English Dictionary does not, as far as I can see online). However, I have never seen it spelled with the circumflex accent anywhere outside of mathematics. So why is it that so many mathematical authors spell it that way? One explanation would be that the authors are French; however, this does not seem to be empirically true. I have most often seen the spelling used by non-French authors. Another possible explanation is that the word "r\^ole" has a technical meaning that differentiates it from "role". However, I can't imagine what it would be. Maybe this habit has been passed on for generations. Can it perhaps be traced back to a misspelling in some influential article? -- Peter From rrosebru@mta.ca Tue Oct 9 22:00:10 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Oct 2007 22:00:10 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IfPlZ-0000qb-N0 for categories-list@mta.ca; Tue, 09 Oct 2007 21:49:37 -0300 Date: Tue, 9 Oct 2007 00:41:19 -0400 From: "Saul Youssef" Subject: categories: Re: Help! To: "Categories list" MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 42 Speaking of videos, I think that this one could be great for motivating students to learn about categories http://claymath.msri.org/voevodsky2002.mov Besides making some strong statements about the importance of categories in the middle of the talk, it's all related to things that undergraduates know about or are about to learn. - Saul On 10/8/07, Toby Bartels wrote: > Vaughan Pratt wrote at last part: > > >In case You-tube > >ever has a video on triples you should probably mention any synonyms for > >"triple" so the students can find the video. > > YouTube has a series of 5 video on triples under the name "monads": > < http://www.youtube.com/results?search_query=monads&search=Search >, > among others by the Catsters < http://www.youtube.com/user/TheCatsters >. > > > --Toby > > > > From rrosebru@mta.ca Tue Oct 9 22:00:10 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Oct 2007 22:00:10 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IfPmY-0000yj-9P for categories-list@mta.ca; Tue, 09 Oct 2007 21:50:38 -0300 Date: Tue, 09 Oct 2007 09:34:19 +0200 From: Lutz Schroeder MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: locally cartesian closed categories Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 43 > I agree with Jean Benabou, Fred Linton and Vaughan Pratt that the > definition of a locally cartesian closed category should NOT require > a terminal object. =20 [...] > I confess that I'm a bit surprised to find that the consensus agrees > with me, so to set matters straight I should also point out that my > argument applies equally to elementary toposes and other familiar > structures of categorical logic. Such as cartesian closed categories, for instance. I would like to take the opportunity to point to my paper "Life without the terminal type" in CSL 2001, where I prove that every "almost" cartesian category, i.e. one without a terminal object, extends uniquely to a cartesian closed category with terminal object. There is also a similar result for toposes; the wording is not quite as straightforward as for cartesian closed categories, as one has to formulate (say) the definition of a subobject classifier without reference to a global element True. I recall having thought about locally cartesian closed categories as well, but I do not think I really got anywhere (and actually I just see there's a remark in the paper that says as much). Lutz Schr=F6der --=20 ------------------------------------------------------------------ PD Dr. Lutz Schr=F6der office @ Universit=E4t Bremen: Senior Researcher Cartesium 2.051 Safe and Secure Cognitive Systems Enrique-Schmidt-Str. 5 DFKI-Lab Bremen FB3 Mathematik - Informatik Robert-Hooke-Str. 5 Universit=E4t Bremen D-28359 Bremen P.O. Box 330 440 D-28334 Bremen phone: (+49) 421-218-64216 Fax: (+49) 421-218-9864216 mail: Lutz.Schroeder@dfki,de www.dfki.de/sks/staff/lschrode ------------------------------------------------------------------ ------------------------------------------------------------- Deutsches Forschungszentrum f=FCr K=FCnstliche Intelligenz GmbH Firmensitz: Trippstadter Strasse 122, D-67663 Kaiserslautern Gesch=E4ftsf=FChrung: Prof. Dr. Dr. h.c. mult. Wolfgang Wahlster (Vorsitzender) Dr. Walter Olthoff Vorsitzender des Aufsichtsrats: Prof. Dr. h.c. Hans A. Aukes Amtsgericht Kaiserslautern, HRB 2313 ------------------------------------------------------------- From rrosebru@mta.ca Tue Oct 9 22:00:10 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Oct 2007 22:00:10 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IfPo9-00018P-G4 for categories-list@mta.ca; Tue, 09 Oct 2007 21:52:17 -0300 Date: Tue, 09 Oct 2007 10:31:42 +0100 From: Steve Vickers MIME-Version: 1.0 To: categories list Subject: categories: Re: What is the right abstract definition of "connected"? Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 44 Vaughan Pratt wrote: > I'd like to say that "connected" is defined on objects of any category C > having an object 1+1 (coproduct of two final objects). X is connected > just when C(X,1+1) <= 2. Dear Vaughan, There's a big reason (there are also some little reasons, but I'll mention them later) why this doesn't match some accepted categorical definitions, and it's to do with the elements of C(1, 1+1). The topological condition is often stated differently: that every map X -> 1+1 factors via 1. Thus C(X,1+1) <= C(1,1+1). I think in most contexts you would want to say that, if anything is connected, 1 is, but you can easily find C(1,1+1) > 2. A simple example is with C = Set^2, where C(1,1+1) = 4 (two coproduct injections, and two more mixed morphisms). Then with this C, the alternative definition gives a useful notion of "fibrewise connectedness" for spaces over 2 and it's really just connectedness in the internal mathematics of (the topos) Set^2. Your definition is external. I would say don't persevere with your definition unless you really don't mind if 1 is disconnected. The different definition of "every map to 1+1 factors via 1" has been quite successful. That was the big reason. The little reasons I alluded to are that it is often useful to require every map to 0 also to factor via 1. That excludes 0 itself from connectedness. This is similar to saying 1 is not prime. Once you have the 0 and 2 cases for X, then for every finite n (= 1 + ... + 1) you have all maps X -> n factor via 1 - at least, if coproduct is well enough behaved w.r.t. limits. In constructive locale theory the standard definition is stronger and requires that for every discrete I, every map X -> I must factor via 1. This allows "infinite n". (Classically this can be deduced from the 0 and 2 cases.) All the best, Steve. From rrosebru@mta.ca Tue Oct 9 22:00:10 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Oct 2007 22:00:10 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IfPqQ-0001MR-VR for categories-list@mta.ca; Tue, 09 Oct 2007 21:54:39 -0300 From: "Jonathon Funk" To: "categories list" Subject: categories: Re: What is the right abstract definition of "connected"? Date: Tue, 9 Oct 2007 10:43:12 -0400 MIME-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 45 One suggestion is to say that an object X in a category C (with products) is connected relative to a functor F:B-->C if passing from maps m:b-->b' in B to maps XxF(b)-->F(b') (by composing the projection XxF(b)-->F(b) with F(m) ) is a bijection for every b,b' (or possibly just onto, not bijection, could be stipulated, but I don't know how inappropriate that would be). If pullbacks exist X*: C-->C/X, then this is equivalent to X*F full and faithful (or just full). If say b=1=terminal of B (and F(1)=1), then it is as if to say that if X is connected (relative to F), then elements of any b' are in bijection with (or at least onto) maps X --> F(b'): every such map is thus `constant'. For example, in this sense we may speak of a connected object X in a topos E-->S relative to Delta: S--->E. Jonathon ----- Original Message ----- From: "Vaughan Pratt" To: "categories list" Sent: Monday, October 08, 2007 4:18 PM Subject: categories: What is the right abstract definition of "connected"? > I'd like to say that "connected" is defined on objects of any category C > having an object 1+1 (coproduct of two final objects). X is connected > just when C(X,1+1) <= 2. > > If this definition appears in print somewhere I can just cite it. If > not is there a better or more standard generally applicable definition I > can use? > > If C(X,1+1) = 2 is citable but not <= 2, have the proponents of =2 > taken into account that no Boolean algebra is connected according to the > =2 definition? This is because 1+1 ~ 1 in Bool, CABA, DLat, StoneDLat, > etc. (dual to 0x0 ~ 0 in Set, Pos, etc.), forcing C(X,1+1) = 1. Boolean > algebras and distributive lattices fail the =2 test not because they are > disconnected in any natural sense but rather because they are > hyperconnected. It seems unreasonable to say that hyperconnected > objects are not connected. > > There is also the question of the object of connected components of an > object. In Set and Grph, if X has k connected components then C(X,1+1) > = 2^k for all X, a set (C being ordinary, i.e. enriched in Set). In > Stone (Stone spaces) however this only holds for finite X, with k = X. > For infinite X Stone(X,1+1) is the set of clopen sets of X, which can be > countably infinite and hence not 2^k for any k. > > If we read 2^k as Stone(k,2), taking k = X and 2 the Sierpinski space > this doesn't help. However Stone(k,1+1) is ideal: instead of treating > the object of connected components of a Stone space k = X as a set we > can treat them as a Boolean algebra, namely that of the clopen sets of X. > > These examples are worth bearing in mind when considering the > appropriate general definition of number of connected components of an > object, and whether even to treat it as a number (cardinal) or a more > general object. > > Connectedness seems somehow more basic than finiteness because we can > easily draw examples of connected and disconnected objects, whereas it > requires a vivid imagination to see the boundary between finite and > infinite objects one might try to draw on paper. > > This motivates making connectedness prior to finiteness. > > Another familiar and easily visualized notion with small examples is > that of path. Define a *path* to be a connected directed graph having > one vertex each of degree (0,1) and (1,0), and all others (1,1). (The > degree (m,n) specifies the in-degree as m and the out-degree as n.) > > We can then define a finite set to be one in bijection with the set of > vertices of some path. This seems more natural than defining it to be > one such that every injection on itself is a surjection, because there > are a lot of injections to worry about and how do you convince yourself > that surjective injections don't kick in until omega? > > Those who are already wedded to some other definition of finite will > want to check that this path-based definition draws the boundary in the > same place as theirs. For what definitions of "finite" can this not be > shown? And are any of them more palatable than the path-based definition? > > Vaughan > > > From rrosebru@mta.ca Tue Oct 9 22:00:10 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 09 Oct 2007 22:00:10 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IfPtn-0001lv-4D for categories-list@mta.ca; Tue, 09 Oct 2007 21:58:07 -0300 Mime-Version: 1.0 (Apple Message framework v752.3) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit From: Ross Street Subject: categories: Re: Help! Date: Wed, 10 Oct 2007 08:33:57 +1000 To: Categories Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 46 Dear Mike Some categories that are easily described (even to talented high school students) are: the category fun of functions (where objects are natural numbers and morphisms are functions); the category mat of matrices (again the objects are natural numbers); the category brd of braids; and, the category tang of tangles. There are enough functors amongst these to be interesting. They are all monoidal categories. One can try to discuss other structure the categories have in common so that strong monoidal functors (although I probably wouldn't introduce too much such terminology) preserve it. For example, duals in mat and tang; trace and braid closure; etc. One could try to show how the specialized, seemingly ad hoc Reidemeister moves translate naturally into the braided monoidal setting. A hint about how the "new" (mid 1980s) polynomial link invariants come from a functor tang --> mat might be of interest. Best wishes, Ross On 05/10/2007, at 10:52 PM, Michael Barr wrote: > What would you say to an undergraduate math club about categories? > I have > been thinking about it, but I am not sure what to say. Talk about > cohomology, which is what motivated E-M? I don't think so. Talk > about > dual spaces of finite-dimensional vector spaces? Maybe, but then > what? From rrosebru@mta.ca Wed Oct 10 17:11:30 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 10 Oct 2007 17:11:30 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ifhod-0006Ai-VK for categories-list@mta.ca; Wed, 10 Oct 2007 17:06:00 -0300 Mime-Version: 1.0 (Apple Message framework v752.3) Content-Type: text/plain; charset=ISO-8859-1; delsp=yes; format=flowed Content-Transfer-Encoding: quoted-printable From: Ross Street Subject: categories: Re: "role" vs. "r\^ole" Date: Wed, 10 Oct 2007 16:34:47 +1000 To: Categories Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 47 Dear Peter With questions like this, we naturally miss Max's expertise. Allow me a few ramblings. While you did take a nice sample of examples from different English speaking countries, they were all from newspapers. In Australia, newspapers have long used the American spellings; they used color, labor, . . . long before other Australians. I found this strange as a kid. In fact, I believe my old Collins Australian Dictionary would have given "r\^ole". The Macquarie Dictionary gives "r\^ole" as a second usage as American influences are growing. The English word, as we know, comes from the French "r\^ole" for roll of paper on which an actor's part was written. It was quite legitimate for Webster to pin down American spelling before it had stabilized in =20= England and before the fashion in England favoured the French spellings such as "programme". I suspect that "r\^ole" was part of that revision. (Funnily, the French at the same time favoured English words and names such as Edith.) So I don't think it is in mathematics especially, except that we are =20 a bit conservative when it comes to language. "Shew" was in my mathematics text books as an undergraduate, and not in any other texts. The typewriter, computer and mobile/cellular phone have also helped eliminate accents (and, unfortunately, apostrophes). Who wants to look at ? Or <=FC> or <=E9> in the way my mailer transforms = it to yours? But I do want to preserve the distinctions: between and , , , and . End of ramble. Ross On 10/10/2007, at 7:04 AM, Peter Selinger wrote: > Does anyone know why it is common, in papers on logic, semantics, and > category theory, to spell the word "role" the French way, i.e., with a > circumflex accent? I am taking about the idiom "to play a role", as > in, "in this definition, x and y play symmetric r\^oles". Sometimes it > is also used as in "the r\^ole of x is ...". From rrosebru@mta.ca Wed Oct 10 17:11:30 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 10 Oct 2007 17:11:30 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IfhsG-0006iV-30 for categories-list@mta.ca; Wed, 10 Oct 2007 17:09:44 -0300 Subject: categories: errata From: Eduardo Dubuc Date: Wed, 10 Oct 2007 10:58:41 -0300 (ART) To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 48 in my posting: "this is essentially the difference between filterness and cofilterness, with all what it means" it should be: "this is essentially the difference between filterness and pseudofilterness, with all what it means" From rrosebru@mta.ca Wed Oct 10 17:11:30 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 10 Oct 2007 17:11:30 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IfhrW-0006au-Oh for categories-list@mta.ca; Wed, 10 Oct 2007 17:08:58 -0300 From: "Marta Bunge" To: categories@mta.ca Subject: categories: RE: What is the right abstract definition of "connected"? Date: Wed, 10 Oct 2007 08:00:15 -0400 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 49 Dear Vaugham, >I'd like to say that "connected" is defined on objects of any category C >having an object 1+1 (coproduct of two final objects). X is connected >just when C(X,1+1) <= 2. > >If this definition appears in print somewhere I can just cite it. If >not is there a better or more standard generally applicable definition I >can use? In Categories and Alligators (1.733, pg 124), an object in a pre-logos is called CONNECTED if it has exactly two complemented subobjects. They observe that in Sh(Y), the terminator is connected iff Y is a connected space. A PRE-LOGOS (pag 98) is a regular category in which Sub(A) is a lattice (not just a semi-lattice) for each A, and in which f#:Sub(B)---> Sub(A) is a lattice homomorphism for each f:A--->B. For a Grothendieck topos e:E---> S (over an arbitrary base S), this definition admits a generalization with "complemented subobject" replaced by "definable subobject", that is, subobjects classified by which, in case S is Boolean, agrees with the Freyd-Scedrov definition. I do not know if this is the sort of abstraction you want. Now for something (not) entirely different: A related notion to the one above is the notion of "abstractly (exclusively) unary" introduced in my thesis (Categories of Set-Valued Functors, University of Pennsylvania, 1966) as part of the definition of an "atom". An object A in a "regular category" X (in the sense of my thesis, which, modulo the stability assumptions is the same as Barr exact) is "abstractly (exclusively) unary" if every A---> \Sum {X_i} in C factors through one (and only one) injection. (The difference with connected is that arbitrary coproducts must be considered and, unlike what I assert in Proposition 11.8, finiteness does not imply this --incorrect use of Zorn's lemma. ) An object A is an "atom" in a "regular category" X if HOM(A,-):X--->Set preserves colimits, thus also the coproducts which exist in X. In particular, A is abstractly (exclusively) unary. More in particular, every A---> B + C factors uniquely trhough one of the injections. The latter is itself equivalent in this context to every A---> 1 + 1 factors uniquely through one of the injections. Just for completeness I state what is shown in my thesis. A "regular" category X is said to be "atomic" is the class of atoms in it is a set and is generating for X. (The funny thing is that almost all the terminology from my thesis was subsequently abandoned -- "atom" was relaced by "A.T.O." (provided exponentiation exists), and "atomic" had a quite different meaning. ) In any case, my theorem reads (all terminology as in my thesis): THM. (Characterization theorem) Let X be any cocomplete atomic regular category. Then there exists a small category C and a functor X--> S^{C^op} which is an equivalence of categories. Conversely every category of set-valued functors S^{C^op} is cocomplete regular atomic. Note: the terminology introduced in my thesis was motivated by the intended theorem which is of the sort "every complete atomic Boolean algebra is isomorphic to a field of sets" (meaning the "field" of all subsets of its set of atoms). There is no published version of my thesis except for microfilms something. The relative version (relative to a monoidal category V) of this characterization theorem is published in Marta Bunge, Relatived Functor Categories and Categories of Algebras, J. Algebra 11 (1), January 1969, 63-101 (communicated by Saunders MacLane). I am sure that I have expanded way more than you would have wanted. Apologies are in order. Cordially, Marta From rrosebru@mta.ca Wed Oct 10 17:11:31 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 10 Oct 2007 17:11:31 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ifhp9-0006FW-Vi for categories-list@mta.ca; Wed, 10 Oct 2007 17:06:32 -0300 Date: Wed, 10 Oct 2007 09:27:29 +0100 (BST) From: Dusko Pavlovic To: Categories List Subject: categories: Re: "role" vs. "r\^ole" MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 50 yes, that is an interesting question. in fact, i have a similar question about the words star and dagger. although they are perfectly acceptable english words on their own, in the context of categories we write *-autonomous and %-compact, even in the titles, where star-autonomous and dagger-compact would say the same, just look easier to pronounce. but then again, with my name and surname gaining and losing accents as i go, maybe i should not ask such questions. -- du$ko On Tue, 9 Oct 2007, Peter Selinger wrote: > Hi everybody, > > here is a frivolous question only tangentially related to category > theory. > > Does anyone know why it is common, in papers on logic, semantics, and > category theory, to spell the word "role" the French way, i.e., with a > circumflex accent? I am taking about the idiom "to play a role", as > in, "in this definition, x and y play symmetric r\^oles". Sometimes it > is also used as in "the r\^ole of x is ...". > > As far as I can tell, the accented spelling is a strange ideosyncrasy, > given that the word "role", without the accent, is a perfectly > acceptable, and very common, English word. Here are some examples I > collected a few years ago: > > How big a role did politics play? -Los Angeles Times, March 27, 2002 > Huge bomb could play role in Iraq. -The Guardian, March 13, 2003 > Australia intends to play a role in [...] Iraq, The Australian, 4/15/2003 > A movie in which Nicole Kidman could play the lead role -Business Times, 1/16/3 > Genetics play a big role in your health. -Citizen-Times.com, April 11, 2003 > Linux prepares to play broader role in embedded systems. -EETimes, 6/11/2001 > The UN would play a central role in running the country. -Guardian, 4/10/2003 > His role is to lead the paddlers through the race -Waterfront News, Oct 2007 > How oil plays a role in an invasion of Iraq. -YellowTimes.org, Jan 22, 2003 > > I realize that Merriam Webster's Dictionary allows "r\^ole" as an > alternate spelling (the Oxford English Dictionary does not, as far as > I can see online). However, I have never seen it spelled with the > circumflex accent anywhere outside of mathematics. > > So why is it that so many mathematical authors spell it that way? One > explanation would be that the authors are French; however, this does > not seem to be empirically true. I have most often seen the spelling > used by non-French authors. Another possible explanation is that the > word "r\^ole" has a technical meaning that differentiates it from > "role". However, I can't imagine what it would be. > > Maybe this habit has been passed on for generations. Can it perhaps be > traced back to a misspelling in some influential article? > > -- Peter > > > From rrosebru@mta.ca Wed Oct 10 17:11:31 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 10 Oct 2007 17:11:31 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IfhnC-0005uq-E7 for categories-list@mta.ca; Wed, 10 Oct 2007 17:04:30 -0300 Date: Tue, 9 Oct 2007 21:31:14 -0400 (EDT) From: Michael Barr To: Categories List Subject: categories: Re: "role" vs. "r\^ole" MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 51 It is, IMHO, a pure affectation. There are a few, very few, English works that might be improved with an accent (e.g the name "Andre", words like "preempt" and a handful of others), but "role" is certainly not one of them. Another affectation is using "topoi" as the plural of topos. If you insist on that, you should use the genetive of "of topos" and the accusative when it is the direct object--not to mention the vocative when addressing a topos. Michael On Tue, 9 Oct 2007, Peter Selinger wrote: > Hi everybody, > > here is a frivolous question only tangentially related to category > theory. > > Does anyone know why it is common, in papers on logic, semantics, and > category theory, to spell the word "role" the French way, i.e., with a > circumflex accent? I am taking about the idiom "to play a role", as > in, "in this definition, x and y play symmetric r\^oles". Sometimes it > is also used as in "the r\^ole of x is ...". > > As far as I can tell, the accented spelling is a strange ideosyncrasy, > given that the word "role", without the accent, is a perfectly > acceptable, and very common, English word. Here are some examples I > collected a few years ago: > > How big a role did politics play? -Los Angeles Times, March 27, 2002 > Huge bomb could play role in Iraq. -The Guardian, March 13, 2003 > Australia intends to play a role in [...] Iraq, The Australian, 4/15/2003 > A movie in which Nicole Kidman could play the lead role -Business Times, 1/16/3 > Genetics play a big role in your health. -Citizen-Times.com, April 11, 2003 > Linux prepares to play broader role in embedded systems. -EETimes, 6/11/2001 > The UN would play a central role in running the country. -Guardian, 4/10/2003 > His role is to lead the paddlers through the race -Waterfront News, Oct 2007 > How oil plays a role in an invasion of Iraq. -YellowTimes.org, Jan 22, 2003 > > I realize that Merriam Webster's Dictionary allows "r\^ole" as an > alternate spelling (the Oxford English Dictionary does not, as far as > I can see online). However, I have never seen it spelled with the > circumflex accent anywhere outside of mathematics. > > So why is it that so many mathematical authors spell it that way? One > explanation would be that the authors are French; however, this does > not seem to be empirically true. I have most often seen the spelling > used by non-French authors. Another possible explanation is that the > word "r\^ole" has a technical meaning that differentiates it from > "role". However, I can't imagine what it would be. > > Maybe this habit has been passed on for generations. Can it perhaps be > traced back to a misspelling in some influential article? > > -- Peter > > From rrosebru@mta.ca Wed Oct 10 17:11:31 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 10 Oct 2007 17:11:31 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ifhpr-0006MO-3S for categories-list@mta.ca; Wed, 10 Oct 2007 17:07:15 -0300 Date: Wed, 10 Oct 2007 10:17:24 +0100 From: Steve Vickers MIME-Version: 1.0 To: Categories List Subject: categories: Re: "role" vs. "r\^ole" Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 52 Dear Peter, Fowler (I have the 1965 edition) says, "there being no other word 'role' from which it has to be kept distinct, both the italics and the accent might well be abandoned." He also refers (under an article on 'morale') to the "sanctity of the French form". The word comes from French, and French gives it the accent, and some like to display their knowledge of this fact. But Fowler's argument is that English is not obliged to keep the French form. Returning to category theory, and topos theory in particular, I can't resist also quoting his guidance on Latin plurals (which surely must apply even more to Greek): "All that can be safely said [regarding whether to prefer or avoid the Latin form] is that there is a tendency to abandon the Latin plurals, and that, when one is really in doubt which to use, the English form should be given the preference." Regards, Steve. Peter Selinger wrote: > Hi everybody, > > here is a frivolous question only tangentially related to category > theory. > > Does anyone know why it is common, in papers on logic, semantics, and > category theory, to spell the word "role" the French way, i.e., with a > circumflex accent? I am taking about the idiom "to play a role", as > in, "in this definition, x and y play symmetric r\^oles". Sometimes it > is also used as in "the r\^ole of x is ...". > > As far as I can tell, the accented spelling is a strange ideosyncrasy, > given that the word "role", without the accent, is a perfectly > acceptable, and very common, English word. Here are some examples I > collected a few years ago: > > How big a role did politics play? -Los Angeles Times, March 27, 2002 > Huge bomb could play role in Iraq. -The Guardian, March 13, 2003 > Australia intends to play a role in [...] Iraq, The Australian, 4/15/2003 > A movie in which Nicole Kidman could play the lead role -Business Times, 1/16/3 > Genetics play a big role in your health. -Citizen-Times.com, April 11, 2003 > Linux prepares to play broader role in embedded systems. -EETimes, 6/11/2001 > The UN would play a central role in running the country. -Guardian, 4/10/2003 > His role is to lead the paddlers through the race -Waterfront News, Oct 2007 > How oil plays a role in an invasion of Iraq. -YellowTimes.org, Jan 22, 2003 > > I realize that Merriam Webster's Dictionary allows "r\^ole" as an > alternate spelling (the Oxford English Dictionary does not, as far as > I can see online). However, I have never seen it spelled with the > circumflex accent anywhere outside of mathematics. > > So why is it that so many mathematical authors spell it that way? One > explanation would be that the authors are French; however, this does > not seem to be empirically true. I have most often seen the spelling > used by non-French authors. Another possible explanation is that the > word "r\^ole" has a technical meaning that differentiates it from > "role". However, I can't imagine what it would be. > > Maybe this habit has been passed on for generations. Can it perhaps be > traced back to a misspelling in some influential article? > > -- Peter > > From rrosebru@mta.ca Wed Oct 10 17:11:31 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 10 Oct 2007 17:11:31 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ifhny-00063A-N4 for categories-list@mta.ca; Wed, 10 Oct 2007 17:05:18 -0300 Date: Wed, 10 Oct 2007 00:42:21 -0400 From: "Fred E.J. Linton" To: "Categories List" Subject: categories: Re: "role" vs. "r\^ole" Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 53 selinger@mathstat.dal.ca (Peter Selinger) asked: > Subject: categories: "role" vs. "r\^ole" > > Does anyone know why it is common, in papers on logic, semantics, and > category theory, to spell the word "role" the French way, i.e., with a > circumflex accent? ... Fowler, in his Modern English Usage, has this to say under the heading Role, rôle: "Though the word is etymologically the same as roll, = meaning the roll of MS. that contained an actor's part, the differentiation is too useful to be sacrificed by spelling always roll. But, there being no other word role from which it has to be kept distinct, both the italics and the accent might well be abandoned. As to the sanctity of the French form, see MORALE." = And, under Morale, Fowler begins: "Is a combination of pandantry and Gallicism to bully us into ... ? ... The right course is to ... abstain from the = French ... , of which we have no need." > As far as I can tell, the accented spelling is a strange ideosyncrasy, > given that the word "role", without the accent, is a perfectly > acceptable, and very common, English word. = > ... > I realize that Merriam Webster's Dictionary allows "r\^ole" as an > alternate spelling (the Oxford English Dictionary does not, as far as > I can see online). = > ... > Maybe this habit has been passed on for generations. Can it perhaps be > traced back to a misspelling in some influential article? Fowler would call it another result of "pedantry with French words." From rrosebru@mta.ca Wed Oct 10 17:11:31 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 10 Oct 2007 17:11:31 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ifhtn-0006y7-Ra for categories-list@mta.ca; Wed, 10 Oct 2007 17:11:19 -0300 Date: Wed, 10 Oct 2007 12:36:23 -0500 From: Alan Jeffrey MIME-Version: 1.0 To: categories list Subject: categories: Re: Does Bind beat Kleisli in Hilbert space? Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 54 I appreciate the long-term vision of your VCs. For your Haskell engineers, isn't this exactly what arrows are designed to do? For your category theorists, isn't this exactly what a premonoidal category or a Freyd category are designed to do? In both cases, it seems that the mismatch is due to the monadic requirement that thunks be first-class citizens: this results in types such as T(T(X)) existing, which seem to be at the root of your problem with overflow exceptions. Of course, premonoidal categories are not as prevalent in the industry as monads are, and the arrows API is not as heavily adopted as the monads API, so you may have problems interfacing to the large existing monadic code-base. I'm sure your leadership team of strategic visionaries will see the value-add. Alan. John Hughes, Generalising Monads to Arrows, in Science of Computer Programming 37, pp67-111, May 2000. John Power and Edmund Robinson. Premonoidal Categories and Notions of Computation, in Mathematical Structures in Computer Science 7(5):453-468, 1997. John Power and Hayo Thielecke. Closed Freyd- and kappa-categories, ICALP'99, LNCS 1644, pp 625-634, Springer, 1999. Vaughan Pratt wrote: > Project: Operation QCvac > Sensitivity level: Black hole > Situation report: Unanticipated overflow exception in a monad > Reporting analyst: Vaughan Pratt > Project status: On hold pending resolution > Action item: Solicit qualified expert opinion > Date: October 7, 2007 > > Situation summary. We're working on a quantum computer in anticipation > of a Request for Proposal (RFP) for the next One Laptop Per Child (OLPC) > computer for a value of "next" that is acceptable to our venture > capitalists (VCs) yet feasible for our engineers. > > To be sure of not being out-competed we've assembled a crack team of > physicists from Fermilab to get the physics right, category theorists > from Fairfield, Iowa to design the linear algebra implementation, > electrical engineers (EEs) from Silicon Valley to build the machine, > Haskell programmers from Glasgow to implement the ideal third-party > value-add software environment, and marketers from Boston to understand > the market's needs and tastes. > > Marketing feels we have to be able to offer lots of storage (qubits). > The physicists said no problem, they work in separable (countably > dimensioned) Hilbert space all the time. (You see how physics > works---physics is scale-invariant, what's good for describing the > universe is good for describing computers.) Marketing said great, > countably infinite storage will make us unbeatable, even Google will > want one. > > Marketing wants to pitch the reliability of our machine. The category > theorists said no problem, on their previous consulting job, D-Wave's > 16-qubit quantum computer, they'd represented linear algebra as God's > own monad, a monoid object (T,mu,eta) of Set^Set implementing matrix > multiplication via the Kleisli construction. They took the functor T(X) > to be the set C^X of X-dimensional almost-everywhere-zero complex > vectors with T(f:X->Y): C^X --> C^Y sending v: C^X to the vector u: C^Y > describable as starting with u = 0 and adding v_x to u_{f(x)} for all x > in X, the multiplication mu_X: C^(C^X) --> C^X to send V: C^(C^X) to u: > C^X with u_x the sum of V_v * v_x over all v in C^X (dot product), and > the unit eta_X: X --> C^X as eta_X(x)(y) = (x=y) (the unit vectors). It > worked like a charm. (You see how category theory works in > computers---everything is an adjunction, or was in the 1970s, nowadays > it's all done with monads.) > > The EEs expressed concern about the ambitious number of qubits. The > category theorists reassured them that infinity held no fears for them > as the infinite set C^(2^16) had worked fine in the D-Wave machine since > mu only encountered finitely many nonzero values when summing over the > domain C^(2^16) of T(f): C^(C(2^16)) --> C(2^16). The physicists > reassured them that linear algebra lifted reliably to infinite > dimensions provided the vectors were kept square summable as confirmed > by a vast body of experimental evidence, so even though the sums would > now be infinite they would still converge. (You see how systems > analysis works---if monads and square-summability each coordinate well > with the world they must coordinate well with one another.) > ... From rrosebru@mta.ca Thu Oct 11 12:38:22 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Oct 2007 12:38:22 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ifzyh-0002kn-8j for categories-list@mta.ca; Thu, 11 Oct 2007 12:29:35 -0300 Date: Wed, 10 Oct 2007 23:43:55 -0400 From: "Fred E.J. Linton" To: Categories List Subject: categories: Re: "role" vs. "r\^ole" Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 55 Steve Vickers, aka s.j.vickers@cs.bham.ac.uk , notes > Fowler (I have the 1965 edition) says, "there being no other word 'role= ' > from which it has to be kept distinct, both the italics and the accent > might well be abandoned." Same words almost exactly, but for the "and" above taking the form of an = "&", in my 1952 reprint of the 1937 (twice corrected) version of the 1911 edit= ion. Fowler takes a dim view of -- one might even say, rails against -- the = use of French words in English: his article "French words" begins, "Display of superior knowledge is as great a vulgarity as display of superior wealth -- greater, indeed, inasmuch as knowledge should tend = more definitely than wealth towards discretion & good manners." Salut :-) , -- Fred From rrosebru@mta.ca Thu Oct 11 12:38:23 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Oct 2007 12:38:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ifzwz-0002Wm-Uv for categories-list@mta.ca; Thu, 11 Oct 2007 12:27:50 -0300 Date: Wed, 10 Oct 2007 19:04:08 -0700 From: Toby Bartels To: Categories List Subject: categories: Re: "role" vs. "r\^ole" MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 56 Michael Barr wrote in part: >Another affectation is using "topoi" as the plural of topos. If >you insist on that, you should use the genetive of "of topos" and the >accusative when it is the direct object--not to mention the vocative when >addressing a topos. That's not really fair; there's a long history in English, when adopting a foreign noun, of adopting the foreign plural as well (then switching to an "-s" plural when the noun becomes less foreign, or occasionally using the plural form only as a collective noun). However, this practice uses only one case, usually nominative. So by saying "topoi", one pretends that "topos" is a real Greek word (in this sense) and that it's still a foreign word with a foreign plural. This pretence is an affectation, certainly, but it is complete in itself. --Toby From rrosebru@mta.ca Thu Oct 11 12:38:23 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Oct 2007 12:38:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ifzu3-00023k-9t for categories-list@mta.ca; Thu, 11 Oct 2007 12:24:47 -0300 Date: Wed, 10 Oct 2007 13:43:38 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories list Subject: categories: Re: What is the right abstract definition of "connected"? Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 57 Steve Vickers wrote: > The topological condition is often stated differently: that every map > X -> 1+1 factors via 1. Thus C(X,1+1) <= C(1,1+1). I think in most > contexts you would want to say that, if anything is connected, 1 is, > but you can easily find C(1,1+1) > 2. Thanks, Steve, this is great. I didn't want to go out on a limb with C(X,1+1) <= 2 (or = 2) if it was buggy, good to know about the C(1,1+1) > 2 problem. This also takes care of my concern about situations where 1+1 = 1, since your definition as stated makes Boolean algebras etc. connected. Presumably my taking the anarchist side (no unity) in the definition of locally cartesian closed obligates me to ask for the right formulation of "connected" in the absence of 1. How about the following? ================================================================= An object of a category is *connected* when its every morphism to a nonempty coproduct factors through an inclusion thereof. ================================================================= This eliminates all assumptions about the category -- if there are no nontrivial coproducts every object is connected by default (any morphism to a trivial coproduct factors through its one inclusion), reasonable when there is no recognizable (by the coproduct test) example of disconnectedness in the category to compare with. It also accomodates: > In constructive locale theory the standard definition is stronger and > requires that for every discrete I, every map X -> I must factor via > 1. This allows "infinite n". with the same benefits - constructive I suppose (how is that judged exactly?), and allows infinite comparisons. If necessary one could qualify "coproduct" with "small" but methodologically it would seem preferable to let such size limits be set by a larger context. The effect of > The little reasons I alluded to are that it is often useful to > require every map to 0 also to factor via 1. That excludes 0 itself > from connectedness. can be had by omitting "nonempty" from the definition. While this might seem a very natural omission, my concern with it is not so much 0 itself as the objects with morphisms to 0, e.g. all Boolean algebras except 1, which this definition would therefore make not connected. Stone spaces being totally disconnected, it just seems plain wrong to have their duals not connected either when they are so obviously connected, like totally (except 1, which is, like, connected but not totally, being dual to the empty Stone space, which is, like, disconnected but not totally). In the geometric duality of points and lines in the plane, two points are disconnected unless they coincide, while two lines are connected unless they are parallel. And an undirected graph and its complement either both contain an N or neither do, and in the latter case you can ask Google the following. Is an N-free graph connected if and only if its complement is disconnected? Google will confirm that it is, no need to click on any of the links it returns. (You may have to read several of Google's "answers" though since Google isn't yet smart enough to just say yes, or even to give the most direct "answer" first.) Graphs with an N are the undirected graph counterpart of the empty Stone space and the one-element Boolean algebra, being neither totally connected nor totally disconnected. Incidentally it's amazing just how many questions Google "knows" the answer to. Like all oracles though it tends to be a little erratic on questions involving future events. Google's staggering R&D budget notwithstanding, asking it whether Hillary will win the election is about as useful as asking the 8-ball: you're way better off asking the people who place sub-Google-sized ($100) bets on such questions. And asking NSF for funding for your research into questions you propose to answer by asking Google has even lower odds than asking Google. Vaughan From rrosebru@mta.ca Thu Oct 11 12:38:23 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Oct 2007 12:38:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ifzue-00029s-FT for categories-list@mta.ca; Thu, 11 Oct 2007 12:25:24 -0300 Date: Wed, 10 Oct 2007 14:11:41 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: Categories List Subject: categories: Re: "role" vs. "r\^ole" Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 58 Michael Barr wrote: > Another affectation is using "topoi" as the plural of topos. If > you insist on that, you should use the genetive of "of topos" and the > accusative when it is the direct object--not to mention the vocative when > addressing a topos. What a great idea. You may have started a movement here. Topos: omicron declension (second) . Singular (one) Dual (two) Plural (many) Nom topos topo topoi Gen topou topoin topon Dat topoi topoin topois Acc topon topo topous Voc tope topo topoi (Source: http://en.wikipedia.org/wiki/Ancient_Greek_grammar ) Be careful when addressing two topo after dinner---if you hail them as topoi they may think you've had one too many. On a related note, anyone know whether topos is masculine or feminine? Ignorance there could get you off to a bad start with two topoin. Vaughan From rrosebru@mta.ca Thu Oct 11 12:38:23 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Oct 2007 12:38:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ifzvz-0002MY-MW for categories-list@mta.ca; Thu, 11 Oct 2007 12:26:47 -0300 Date: Wed, 10 Oct 2007 15:08:09 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: What is the right abstract definition of "connected"? Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 59 Dear Marta and Jonathan, As it turns out I really only needed the definition for categories of directed graphs, where "An object of a category is *connected* when its every morphism to a nonempty coproduct factors through an inclusion thereof" does exactly what I wanted there (if I haven't messed up my generalization of Steve Vickers' definition). This raises the interesting question however of whether the definitions you both mentioned differ from the above in the categories to which they apply, and if so which notion is preferable in those categories and why? What about Cat&Al's Sh(Y) for example? You both may have such examples; if not then I would argue that my definition has the advantages of generality and simplicity. Best, Vaughan Jonathan Funk wrote: > One suggestion is to say that an object X in a category C (with products) is > connected relative to a functor F:B-->C if passing from maps m:b-->b' in B > to maps > XxF(b)-->F(b') (by composing the projection XxF(b)-->F(b) with F(m) ) is a > bijection for every b,b' > (or possibly just onto, not bijection, could be stipulated, but I don't know > how inappropriate that would be). > > If pullbacks exist X*: C-->C/X, then this is equivalent to X*F full and > faithful (or just full). From rrosebru@mta.ca Thu Oct 11 12:38:23 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Oct 2007 12:38:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ifzxj-0002cZ-9F for categories-list@mta.ca; Thu, 11 Oct 2007 12:28:35 -0300 From: Robert L Knighten Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Date: Wed, 10 Oct 2007 20:04:22 -0700 To: Categories List Subject: categories: Re: "role" vs. "r\^ole" Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 60 Steve Vickers writes: > > Returning to category theory, and topos theory in particular, I can't > resist also quoting his guidance on Latin plurals (which surely must > apply even more to Greek): "All that can be safely said [regarding > whether to prefer or avoid the Latin form] is that there is a tendency > to abandon the Latin plurals, and that, when one is really in doubt > which to use, the English form should be given the preference." > Having looked at both a few authorities and some publications outside of category theory where the word is used, I am curious if there is any field other than category theory where the plural of topos is not topoi? -- Bob -- Robert L. Knighten RLK@knighten.org From rrosebru@mta.ca Thu Oct 11 12:38:23 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Oct 2007 12:38:23 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IfzvI-0002GS-Ma for categories-list@mta.ca; Thu, 11 Oct 2007 12:26:04 -0300 Date: Wed, 10 Oct 2007 17:39:22 -0400 (EDT) From: Michael Barr To: Categories List Subject: categories: Re: Re: "role" vs. "r\^ole" MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 61 Let me tell you a (slightly) amusing story. Beno Eckmann warned me that all librarians would hate me if I called it "$*$-Autonomous Categories". I saw his point immediately and had every intention of changing it. But in the process of getting it typed and so on, I just plain forgot. So although I do object to *-autonomous (without the dollar signs that put it on the line), I would have no problem with star-autonomous. On ne saurait penser a tout, as they say on some obscure langauge. Michael On Wed, 10 Oct 2007, Dusko Pavlovic wrote: > yes, that is an interesting question. in fact, i have a similar question > about the words star and dagger. although they are perfectly acceptable > english words on their own, in the context of categories we write > *-autonomous and %-compact, even in the titles, where star-autonomous and > dagger-compact would say the same, just look easier to pronounce. > > but then again, with my name and surname gaining and losing accents as i > go, maybe i should not ask such questions. > > -- du$ko From rrosebru@mta.ca Thu Oct 11 22:24:12 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Oct 2007 22:24:12 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ig9F1-0002JY-Hy for categories-list@mta.ca; Thu, 11 Oct 2007 22:23:03 -0300 Date: Fri, 12 Oct 2007 00:35:36 +0200 From: Bernhard Beckert MIME-Version: 1.0 To: undisclosed-recipients:; Subject: categories: CFP: TAP 2008 - The Second International Conference on Tests and Proof Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 62 TAP 2008 Second International Conference on Tests and Proofs April 9-11, 2008, Prato (near Florence), Italy http://www.uni-koblenz.de/tap2008/ *CALL FOR PAPERS* SCOPE The TAP conference is devoted to the convergence of proofs and tests. It combines ideas from both sides for the advancement of software quality= . To prove the correctness of a program is to demonstrate, through impeccab= le mathematical techniques, that it has no bugs; to test a program is to run= it with the expectation of discovering bugs. The two techniques seem contradictory: if you have proved your program, it's fruitless to comb it= for bugs; and if you are testing it, that is surely a sign that you have give= n up on any hope to prove its correctness. Accordingly, proofs and tests have, since the onset of software engineeri= ng research, been pursued by distinct communities using rather different techniques and tools. And yet the development of both approaches leads to the discovery of comm= on issues and to the realization that each may need the other. The emergence= of model checking has been one of the first signs that contradiction may yie= ld to complementarity, but in the past few years an increasing number of resear= ch efforts have encountered the need for combining proofs and tests, droppin= g earlier dogmatic views of incompatibility and taking instead the best of = what each of these software engineering domains has to offer. How does deduction help testing? How does testing help deduction? How can the combination of testing and deduction increase the reach of b= oth? TOPICS Topics include: - Generation of test data, oracles, or preambles by deductive technique= s such as theorem proving, model checking, symbolic execution, constraint logic programming, etc. - Generation of specifications by deduction - Verification techniques combining proofs and tests - Program proving with the aid of testing techniques - Transfer of concepts from testing to proving (e.g., coverage criteria= ) - Automatic bug finding - Formal frameworks - Tool descriptions and experience reports - Case studies IMPORTANT DATES November 2, 2007: Abstract submission deadline November 9, 2008: Paper submission deadline January 20, 2008: Acceptance notification February 3, 2008: Final version due April 9-11, 2008: Conference SUBMISSIONS Submissions should describe previously unpublished work (completed or in progress), including descriptions of research, tools, and applications. Papers must be formatted following the Springer LNCS guidelines and be at= most 15 pages long. Submission of papers is via EasyChair at http://www.easychair.org/TAP2008= /. The proceedings are planned to be published within Springer's LNCS series. They will be available at the conference. CONFERENCE CHAIR B. Meyer (ETH Zurich, Switzerland) PROGRAM CO-CHAIRS B. Beckert (U of Koblenz, Germany) R. H=E4hnle (Chalmers U of Technology, Sweden) PROGRAMME COMMITTEE B. Aichernig (TU Graz, Austria) M. Butler (U of Southampton, UK) P. Chalin (Concordia U Montreal, Canada) T.Y. Chen (Swinburne U of Technology, Australia) Y. Gurevich (Microsoft Research, USA) D. Hamlet (Portland State U, USA) W. Howden (U of California at San Diego, USA) D. Jackson (MIT, USA) K. Meinke (KTH Stockholm, Sweden) B. Meyer (ETH Zurich, Switzerland) P. M=FCller (Microsoft Research, USA) T. Nipkow (TU M=FCnchen, Germany) A. Polini (U of Camerino, Italy) Robby (Kansas State U, USA) D. Rosenblum (U College London, UK) W. Schulte (Microsoft Research, USA) N. Sharygina (U of Lugano, Switzerland, and CMU, USA) B. Venneri (U of Firenze, Italy) B. Wolff (ETH Zurich, Switzerland) STEERING COMMITTEE Y. Gurevich (Microsoft Research, USA) B. Meyer (ETH Zurich, Switzerland) ORGANIZING COMMITTEE C. Gladisch (U of Koblenz, Germany) P. R=FCmmer (Chalmers U of Technology, Sweden) CONTACT Email: tap2008@uni-koblenz.de Web: http://www.uni-koblenz.de/tap2008/ From rrosebru@mta.ca Thu Oct 11 22:24:12 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Oct 2007 22:24:12 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ig99x-0001qN-A5 for categories-list@mta.ca; Thu, 11 Oct 2007 22:17:49 -0300 From: "Marta Bunge" To: categories@mta.ca Subject: categories: Re: What is the right abstract definition of "connected"? Date: Thu, 11 Oct 2007 14:48:09 -0400 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 63 Dear Vaughan ============================================================== An object of a category is *connected* when its every morphism to a nonempty coproduct factors through an inclusion thereof. ============================================================== Your proposed definition above is precisely the notion of *abstractly unary* from my J.Algebra '69 paper. It was so termed (instead of *connected*) since it does not need a terminal object to state it (precisely your motivation) and since one does not want to restrict to binary coproducts. When there is a terminal object, and when the coproducts considered are just the binary ones, it is enough to consider morphisms into the coproducts 1+1 (as I show in my thesis) and, in that case, it should be simply called *connected*. In another guise, this is the definition of *connected* given in Cats and Alligators, and it is the one directly inspired by topology. I see no reason to change the terminology. In short, your connected objects I have called abstractly unary. They came about in connection with atoms. An object A in a cocomplete (concrete) category E is an *atom* if HOM(A,-):E--->Set preserves colimits. More objectively, if E has exponentiation, Lawvere uses the notion of an *A.T.O.* instead, meaning that the functor (-)^A : E---> E has a right adjoint (the "amazing right adjoint"). I hope this helps, Cordially, Marta From rrosebru@mta.ca Thu Oct 11 22:24:12 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Oct 2007 22:24:12 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ig9Br-00021q-9h for categories-list@mta.ca; Thu, 11 Oct 2007 22:19:47 -0300 MIME-Version: 1.0 Content-Type: text/plain; charset="us-ascii" Content-Transfer-Encoding: quoted-printable Subject: categories: Re: What is the right abstract definition of "connected"? Date: Fri, 12 Oct 2007 09:05:14 +1000 From: "Stephen Lack" To: "categories list" Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 64 Dear Vaughan, Lawvere and Janelidze have each argued for many years (in somewhat different contexts) that notions of connectedness and cohesion should be understood as relative. This impacts on both your questions: how should connectedness be defined, and what sort of answers should be allowed to the question ``how many connected components does X have?'' --- the=20 second question becomes ``what is the codomain of the pi_0 functor?'' Steve Vickers mentioned the example Set^2. He said that the terminal object (1,1) is obviously connected. But it is equally obviously not connected: (1,1)=3D(1,0)+(0,1). The latter point of view comes from thinking of = Set^2 as a Set-topos, where the connected components functor becomes the functor Set^2-->Set given by homming out of (1,1). The former point of view comes from thinking of Set^2 as defined over itself; then, as Steve says, (1,1) becomes almost tautologically connected, since pi_0 is just the identity functor Set^2-->Set^2. If crng is the category of finitely presentable commutative rings with no non-trivial nilpotents, then there is a lovely pi_0:crng^op-->set_f. For in this case every ring R splits as R_1 x R_2 x ... x R_n, where the R_i have no non-trivial idempotents. It is these R_i which are your connected=20 components. For a larger category of commutative rings, you have to expand=20 your notion of connected component to something like Stone spaces.=20 For a locally connected topos E, defined over S, the inverse image functor e^*:S-->E has not just a right adjoint e_* but also a left adjoint e_!, which serves as pi_0. But one can describe just in terms of e_! -| e^* (i.e. without mention of e_*, and without all of the topos structure) the sorts of abstract=20 properties needed for a good pi_0. This is the starting point for Janelidze's Galois theory.=20 If E is infinitarily extensive (small coproducts, which are stable under pullback and disjoint), then a good notion of connectedness of an object X is that=20 the hom-functor E(X,-):E-->Set preserves coproducts. This includes the locally connected topos case, which in turn includes your case of directed graphs.=20 The case of crng is a finitary version.=20 Regards, Steve Lack. From rrosebru@mta.ca Thu Oct 11 22:24:13 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 11 Oct 2007 22:24:13 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ig9E2-0002EJ-5t for categories-list@mta.ca; Thu, 11 Oct 2007 22:22:02 -0300 Date: Fri, 12 Oct 2007 02:43:18 +0200 (CEST) From: Mikael Vejdemo Johansson To: Categories List Subject: categories: Re: "role" vs. "r\^ole" MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 65 [Note from moderator: this may be a good note on which to close this thread.] On Wed, 10 Oct 2007, Vaughan Pratt wrote: > Michael Barr wrote: >> Another affectation is using "topoi" as the plural of topos. If >> you insist on that, you should use the genetive of "of topos" and the >> accusative when it is the direct object--not to mention the vocative when >> addressing a topos. > > What a great idea. You may have started a movement here. > > Topos: omicron declension (second) > > . Singular (one) Dual (two) Plural (many) > Nom topos topo topoi > Gen topou topoin topon > Dat topoi topoin topois > Acc topon topo topous > Voc tope topo topoi > > (Source: http://en.wikipedia.org/wiki/Ancient_Greek_grammar ) > > Be careful when addressing two topo after dinner---if you hail them as > topoi they may think you've had one too many. On a related note, anyone > know whether topos is masculine or feminine? Ignorance there could get > you off to a bad start with two topoin. > Now, now, if we're really embracing the ancient grammar to this extent, then we should drop the prepositions that english uses to accomodate the same semantic space that once was handled by cases. Thus, I would rewrite the above as "Ignorance there could get you off to a bad start topoin." And of course, once we enter this course, what's more natural than letting old english, and even old norse guide our cases throughout? -- Mikael Vejdemo Johansson | To see the world in a grain of sand mik@math.su.se | And heaven in a wild flower | To hold infinity in the palm of your hand | And eternity for an hour From rrosebru@mta.ca Fri Oct 12 10:06:54 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 12 Oct 2007 10:06:54 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IgKAL-000322-Gd for categories-list@mta.ca; Fri, 12 Oct 2007 10:02:57 -0300 Date: Fri, 12 Oct 2007 11:06:43 +0300 (EEST) Subject: categories: PSSL 87 in Patras - Honouring A. Kock, on the occasion of his 70th birthday From: "Panagis Karazeris" To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain;charset=iso-8859-7 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 66 PSSL 87 in Patras - Honouring A. Kock, on the occasion of his 70th birthd= ay The 87th Peripatetic Seminar on Sheaves and Logic will be held in Patras, Greece, the weekend 22-23 March 2008. The meeting will have an extra festive character, honouring Anders Kock on the occasion of his 70th birthday. For that reason an extra session of invited talks will take place on Friday 21 March. A. Joyal, F.W. Lawvere and G. Reyes have accepted to participate in the celebration with plenary talks. People who wish to attend may thus plan for a longer stay than a usual PSSL would imply. The need for a longer stay is propped by the fact that Patras is, for most people in the community, a more remote destination than those in central and western Europe where Peripatetic Seminars were previously held. Please take also into account that the weekend 22-23 March is that of the Catholic Easter (not of the Greek Orthodox one though). Travelling information, possible arrangements for transportation from Athens airport and further details will appear in the webpage www.math.upatras.gr/~pssl87 People who wish to attend may fill out the following registration form an= d submit it to pssl87@math.upatras.gr =3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D REGISTRATION FORM Name: ....... Affiliation: ..... I wish to give a talk: YES/NO Title: ..... Arrival Date: .... Departure Date: .... I wish to book a hotel room: SINGLE/DOUBLE From rrosebru@mta.ca Fri Oct 12 10:06:54 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 12 Oct 2007 10:06:54 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IgK8g-0002qE-Bu for categories-list@mta.ca; Fri, 12 Oct 2007 10:01:14 -0300 Date: Fri, 12 Oct 2007 00:30:47 -0400 From: "Fred E.J. Linton" To: Categories List Subject: categories: Re: "role" vs. "r\^ole" Mime-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 67 Some examples in support of what Toby Bartels = wrote: > ... there's a long history in English, > when adopting a foreign noun, of adopting the foreign plural as well ..= =2E Examples: alumni and alumnae, not alumnuses; simplices, not simplexes; = vertices, not vertexes; phenomena, not phenomenons; data, not datums. [Not that there aren't counterexamples, too.] -- Fred From rrosebru@mta.ca Fri Oct 12 10:06:54 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 12 Oct 2007 10:06:54 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IgK9b-0002x6-Mi for categories-list@mta.ca; Fri, 12 Oct 2007 10:02:11 -0300 Mime-Version: 1.0 (Apple Message framework v624) Content-Type: text/plain; charset=US-ASCII; format=flowed Content-Transfer-Encoding: 7bit From: David Yetter Subject: categories: Re: "role" vs. "r\^ole" Date: Thu, 11 Oct 2007 23:37:15 -0500 To: Categories Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 68 Perhaps all the categorists who insist on toposes instead of topoi will teach their lower division students about coordinate axises and look forward to greeting them when they return to the university as alumnuses? (Naturally if they do this they will omit the diaresis from the second o in coordinate.) (Sorry to be contrarian, but I've always been fond of the Greek plural, and the diaresis on the o in co\"{o}rdinate.) Best Thoughts, David Y. On 10 Oct 2007, at 22:04, Robert L Knighten wrote: > Steve Vickers writes: >> >> Returning to category theory, and topos theory in particular, I can't >> resist also quoting his guidance on Latin plurals (which surely must >> apply even more to Greek): "All that can be safely said [regarding >> whether to prefer or avoid the Latin form] is that there is a tendency >> to abandon the Latin plurals, and that, when one is really in doubt >> which to use, the English form should be given the preference." >> > > Having looked at both a few authorities and some publications outside > of > category theory where the word is used, I am curious if there is any > field > other than category theory where the plural of topos is not topoi? > > -- Bob > > -- > Robert L. Knighten > RLK@knighten.org > From rrosebru@mta.ca Sun Oct 14 18:58:48 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 14 Oct 2007 18:58:48 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IhBJH-0005xE-Pi for categories-list@mta.ca; Sun, 14 Oct 2007 18:47:43 -0300 From: Gaucher Philippe To: categories list Subject: categories: Preprint: Homotopical interpretation of globular complex by multipointed d-space Date: Sun, 14 Oct 2007 19:59:47 +0200 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 69 Dear All, Here is a new preprint: Title: Homotopical interpretation of globular complex by multipointed d-space Abstract: Globular complexes were introduced by E. Goubault and the author to model higher dimensional automata. Globular complexes are topological spaces equipped with a globular decomposition which is the directed analogue of the cellular decomposition of a CW-complex. We prove that there exists a combinatorial model category such that the cellular objects are exactly the globular complexes and such that the homotopy category is equivalent to the homotopy category of flows. The underlying category of this model category is a variant of M. Grandis' notion of d-space over a topological space colimit generated by simplices. This result enables us to understand the relationship between the framework of flows and other works in directed algebraic topology using d-spaces. It also enables us to prove that the underlying homotopy type functor of flows can be interpreted up to equivalences of categories as the total left derived functor of a left Quillen adjoint. Comment: 28 pages, 2 figures Url: http://www.pps.jussieu.fr/~gaucher/Mdtop.ps http://www.pps.jussieu.fr/~gaucher/Mdtop.pdf From rrosebru@mta.ca Sun Oct 14 18:58:48 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 14 Oct 2007 18:58:48 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IhBG9-0005df-7h for categories-list@mta.ca; Sun, 14 Oct 2007 18:44:29 -0300 Mime-Version: 1.0 (Apple Message framework v624) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: <2f1577a52d162bd48837cdfa427d7326@PaulTaylor.EU> From: Paul Taylor Subject: categories: connectedness Date: Fri, 12 Oct 2007 14:53:39 +0100 To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 70 Vaughan Pratt's original enquiry was actually in the context of graph theory (as I suspected at the time, and he subsequently confirmed), but I would like to add something from the point of view of constructive real analysis. First, though, I would like to underline something that Steve Lack (almost) said, namely that the category in which you index your components, and therefore also the one in which you define connectedness, need to be EXTENSIVE, ie their coproducts should be disjoint, and stable under pullback, and the initial object strict. Maybe we've over-done philology recently, but "component" means "putting together", where we expect the parts to cover the whole (coproduct), without overlapping (disjoint), to be distinguishable (like disjoint union, but unlike addition and disjunction). The modern notion of extensivity, in which Steve had a part, captures this idea very neatly. The equivalence between definitions of connectedness based on 1+1 and on X+Y surely depends on stability under pullback, and the requirement that the choice between left and right be unique surely requires disjointness. Maybe a close study of Marta Bunga's work on abstract connectedness would clarify this. Vaughan originally asked about various categories of algebras, and Steve mentioned commutative rings, but quietly turned their arrows around. Stone duality would suggest to me that one should look for connectedness of algebras in their OPPOSITE category of "spaces", which I understand in a generic sense that includes sets, graphs, predomains, locales and affive varieties. Turning to constructive analysis, let me call the categorical definitions above that involve coproducts "binary" and "infinitary classical connectedness". In (almost) traditional topological language, a space X has the binary classical connectedness property if, for any two open subspaces U and V of X, IF they cover and are disjoint and inhabited THEN false. The definition of connectedness that is used in constructive analysis moves one of the hypotheses to the conclusion: IF they cover and are inhabited THEN their intersection is inhabited. From this definition we immediately obtain an APPROXIMATE INTERMEDIATE VALUE THEOREM: IF a function f:X->R on a connected space takes both positive (greater than -epsilon is enough) and negative (less than +epsilon) values, say on inhabited open spaces U and V, then, as U and V cover, they must intersect, ie the function takes values within epsilon of zero. There are well known examples of spaces that pass the classical definition of connectedness, whilst intuitively being made up of two or more parts (for example the graph of sin(1/x) together with the y-axis). Fewer spaces are connected in the constructive sense, but I can't see any examples in which this might fix the classical mis-definition. There are other ways of permuting the hypotheses and conclusions of this definition. In particular, when the space X is compact, the notion of covering it with opens can be internalised using the universal quantifier or necessity operator []. Similarly, if it is overt, habitation can be internalised using the existential quantifier or possibility operator <>. Pushing these conditions across the implication can only be done over an intuitionistic set theory at the cost of double negation. However, in ASD, where open and closed subspaces are related via continuous functions, and not set-theoretic complementation, the Phoa principle allows this switch to be made without the not-not. In the case of a compact overt space such as the interval [0,1], the classical, constructive, compact and over definitions of connectedness agree. For a space that is either not compact or not overt, one of the hypotheses must remain as an equation on the left of |-. Then constructive and overt connectedness agree: U cup V = X |- <>U and <>V implies <>(U cap V) Compact connectedness is U cap V = 0 |- [](U cup V) implies []U or []V. The latter gives rise to another approximate intermediate value theorem: if f:K->R takes values >=0 and <=0 on OCCUPIED subspaces, then its space of zeroes is also occupied. Here, OCCUPIED is the name that I propose for compact spaces whose terminal projection is a proper surjection, just as an INHABITED space is an overt one with an open surjection to 1. An occupied space need not have any points. So far, I have only mentioned BINARY notions of connectedness, but if we want to talk about families of connected COMPONENTS then we must also consider INFINITARY connectedness (as Marta stressed). Here the results for the constructive real line are somewhat surprising. In order to avoid dependent types, I have found it more convenient to discuss infinitary connectedness in terms of equivalence relations. In classical analysis, any OPEN EQUIVALENCE RELATION on [0,1], ie any open subspace of the square that includes the diagonal, is symmetric in it and has the transitivity property, is INDISCRIMINATE - it relates 0 to 1 and indeed any point to any other. This is not the case in Bishop's or Russian Recursive Analysis. There is an open equivalence relation on [0,1] with infinitely many equivalence classes, ie the interval fails the infinite connectedness condition. The quotient by this equivalence relation, ie the space that indexes the components, is discrete but not Hausdorff, ie it admits an equality relation that is not decidable. This one of the reasons why, at variance with many constructive analysts, I believe that the HEINE--BOREL theorem is a necessary part of analysis. In ASD, which obeys Heine--Borel, any open equivalence relation on [0,1] or R is indiscriminate, as in the classical situation, and the line and interval are connected in the infinitary senses. Moreover, any open subspace of R is the disjoint union of countably many open intervals, where each of these words needs careful constructive re-definition. These results are in my paper "A lambda calculus for real analysis", which was presented at CCA 2005 and you can obtain from www.PaulTaylor.EU/ASD I should point out that I am at the moment re-writing part of this paper, to include a "need to know" introduction to continuous lattices, cf my recent posting on this. However, the results that I have discussed above are in the "stable" part of the text. Paul Taylor From rrosebru@mta.ca Sun Oct 14 18:58:48 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 14 Oct 2007 18:58:48 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IhBI4-0005pH-AM for categories-list@mta.ca; Sun, 14 Oct 2007 18:46:28 -0300 Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit From: daniele radicioni Date: Fri, 12 Oct 2007 16:02:26 +0200 To: fomi2008@di.unito.it Subject: categories: [fomi2008] FOMI 2008 - First Announcement and Call for Papers Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 71 FOMI 2008 - 3rd Workshop on Formal Ontologies Meet Industry June 5-6, 2008, Torino, Italy FIRST ANNOUNCEMENT AND CALL FOR PAPERS Conference web site: http://www.fomi2008.di.unito.it This event is jointly organized by: - Laboratory for Applied Ontology, ISTC-CNR, Trento - University of Torino - University of Verona CONFERENCE AIMS FOMI is an international forum where academic researchers and industrial practitioners meet to analyze and discuss issues related to methods, theories, tools and applications based on formal ontologies. There is today wide agreement that knowledge modeling and the semantic dimension of information plays an increasingly central role in networked economy: semantic-based applications are relevant in distributed systems such as networked organizations, organizational networks, and in distributed knowledge management. These knowledge models in industry aim to provide a framework for information and knowledge sharing, reliable information exchange, meaning negotiation and coordination between distinct organizations or among members of the same organization. New tools and applications have been and are being developed in diverse application fields, ranging from business to medicine, from engineering to finance, from law to electronics. All these systems have exploited the theoretical results and the practical experience of previous work. In all cases, it has been shown that formal ontologies play a central role in describing in a common and understandable way the logical and practical features of the application domain. The success of the methodologies associated with knowledge modeling and ontologies led to increased need of a comparison between different approaches and results, with the aim of evaluating the interdependencies between theories and methods of formal ontology and the activities, processes, and needs of enterprise organizations. The FOMI 08 Workshop aims to advance in this direction by bringing together researchers and practitioners interested in ontology application, paying particular attention to the topics listed below. CONFERENCE TOPICS *problems in ontology application:* - practical issues in using ontologies in the enterprise - real cases of successful/unsuccessful use of ontology in business - from legacy systems to the new ontology-driven systems *ontology and business:* - ontology and ontological methodologies in business; - adaptation of ontologies for companies and organizations; - ontology effectiveness and evaluation in business *ontology and enterprise:* - ontology-driven enterprise modeling; - ontology development and change within organizations; - ontology-driven representation of products, services, functionalities, design, processes; *ontology and enterprise knowledge:* - ontologies for the know-how; - ontologies for corporate knowledge; *ontology in practice:* - ontologies for electronic catalogs, e-commerce, e-government; - ontologies for marketing; - ontologies for finance; - ontologies for engineering; - ontologies for medical sciences; *ontology and linguistics:* - ontology-driven linguistic representation in organization knowledge; - linguistic problems in standards and in codification processes; - ontologies and multilingualism in business and organizations PROGRAMME The Scientific Programme will include invited talks, oral presentations, poster and demo presentations, and panels. Submitted papers will be peer-reviewed and selected on the basis of technical quality, relevance of the described experiences (depending on the type of submission), and clarity of the presentation for the workshop. In particular, we insist that papers should be written for a wide audience. Accepted papers will be presented at the workshop, and published as proceedings. Accepted papers will be electronically published on CD and distributed to participants. Following the past edition, a selection of the best papers accepted at the workshop will be considered for publication in the international journal "Applied Ontology" SUBMISSIONS AND DATES * Format - The maximal paper length is 10 pages, excluding title page and bibliography. - Papers must be submitted in PDF format - Detailed instructions can be found in the conference site. * Deadlines: - Paper Submission: January 7, 2008 - Acceptance Notification: March 3, 2008 - Camera Ready: March 31, 2008 CONFERENCE PROGRAMME COMMITTEE Bill Andersen, Ontology Works, USA Peter Clark, Knowledge Systems, Boeing Maths and Computing Technology, USA Matteo Cristani, University of Verona, Italy Roberta Cuel, University of Trento, Italy Roberta Ferrario, Laboratory for Applied Ontology, CNR, Trento, Italy Michael Gruninger, University of Toronto, Canada Nicola Guarino, Laboratory for Applied Ontology, CNR, Trento, Italy Paulo Leitao, Escola Superior de Tecnologia e de Gestao, Polytechnic Institute of Braganca, Portugal Jorge Posada, VICOMTech, Donostia / San Sebastian, Spain Chris Partridge, 42 Objects Limited, BORO Centre Limited, Brunel University, UK Valentina Tamma, University of Liverpool, UK Matthew West, Shell International Petroleum Company Limited, UK From rrosebru@mta.ca Mon Oct 15 10:12:37 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 15 Oct 2007 10:12:37 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IhPf8-0006mO-7y for categories-list@mta.ca; Mon, 15 Oct 2007 10:07:14 -0300 From: "Marta Bunge" To: categories@mta.ca Subject: categories: RE: connectedness Date: Sun, 14 Oct 2007 19:44:37 -0400 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 72 Dear Paul, In the following (private) response to Vaughan, I cleared up a couple of points from my previous posting. I reproduce it here publicly since those points may be relevant to some of the things you wrote. But I really have nothing else to say (at the moment) so no need to reply. Best regards, Marta >From: "Marta Bunge" >Reply-To: marta.bunge@mcgill.ca >To: rrosebrugh@mta.ca >CC: pratt@cs.stanford.edu >Subject: On the connectedness condition >Date: Fri, 12 Oct 2007 06:16:49 -0400 > >Dear Robert, > >I think that I have expanded enough in my response to Vaughan that you >already posted. There was a slight hitch in it, but on the whole is what I >intended to say. I would leave it at that. In any case I am sending this >cc. to Vaughan. > >The hitch is that only in the `at most' part in the definition of >`abstractly exclusively unary >' can one reduce the case to coproducts of 1 (should a terminal exist), but >the `at least' part refers to arbitrary coproducts and does *not* reduce to >coproducts of 1. > > >So, A is `abstractly exclusively unary' if HOM(A,-):E---> SET preserves >coproducts, and it is an `atom' if HOM(A,-):E---> SET preserves colimits. >What Vaughan calls `connected' is what I have called `abstractly unary' >but, more appropriately, `connected' should mean `abstractly exclusively >unary' (the factorization through the injections should be exactly one and >not just at least one). The case of abstractly exclusively unary wrt binary >coproducts of 1 is what Freyd-Scedrov (and all topologists) call connected. > >It would not be inappropriate to equate `connected' with `abstracly >exclusively unary', but not with just `abstractly unary' as Vaughan does. >In other words, = rather than just >, or full and faithful rather than >just full. I think that this was the real issue in Vaughan's question. >This is all there is to it. > > >Best regards, >Marta > >From: Paul Taylor >To: categories@mta.ca >Subject: categories: connectedness >Date: Fri, 12 Oct 2007 14:53:39 +0100 > >Vaughan Pratt's original enquiry was actually in the context of >graph theory (as I suspected at the time, and he subsequently >confirmed), but I would like to add something from the point of >view of constructive real analysis. > >First, though, I would like to underline something that Steve Lack >(almost) said, namely that the category in which you index your >components, and therefore also the one in which you define >connectedness, need to be EXTENSIVE, ie their coproducts should >be disjoint, and stable under pullback, and the initial object strict. > >Maybe we've over-done philology recently, but "component" means >"putting together", where we expect the parts to cover the whole >(coproduct), without overlapping (disjoint), to be distinguishable >(like disjoint union, but unlike addition and disjunction). >The modern notion of extensivity, in which Steve had a part, >captures this idea very neatly. The equivalence between definitions >of connectedness based on 1+1 and on X+Y surely depends on stability >under pullback, and the requirement that the choice between left >and right be unique surely requires disjointness. Maybe a close >study of Marta Bunge's work on abstract connectedness would clarify >this. > >Vaughan originally asked about various categories of algebras, >and Steve mentioned commutative rings, but quietly turned their >arrows around. Stone duality would suggest to me that one should >look for connectedness of algebras in their OPPOSITE category of >"spaces", which I understand in a generic sense that includes >sets, graphs, predomains, locales and affive varieties. > >Turning to constructive analysis, let me call the categorical >definitions above that involve coproducts "binary" and >"infinitary classical connectedness". > >In (almost) traditional topological language, a space X has the >binary classical connectedness property if, for any two open >subspaces U and V of X, > IF they cover and are disjoint and inhabited THEN false. > >The definition of connectedness that is used in constructive >analysis moves one of the hypotheses to the conclusion: > IF they cover and are inhabited THEN their intersection is inhabited. > >From this definition we immediately obtain an APPROXIMATE INTERMEDIATE >VALUE THEOREM: IF a function f:X->R on a connected space takes both >positive (greater than -epsilon is enough) and negative (less than >+epsilon) values, say on inhabited open spaces U and V, then, as >U and V cover, they must intersect, ie the function takes values >within epsilon of zero. > >There are well known examples of spaces that pass the classical >definition of connectedness, whilst intuitively being made up of >two or more parts (for example the graph of sin(1/x) together with >the y-axis). Fewer spaces are connected in the constructive sense, >but I can't see any examples in which this might fix the classical >mis-definition. > >There are other ways of permuting the hypotheses and conclusions of >this definition. In particular, when the space X is compact, the >notion of covering it with opens can be internalised using the >universal quantifier or necessity operator []. Similarly, if it >is overt, habitation can be internalised using the existential >quantifier or possibility operator <>. > >Pushing these conditions across the implication can only be done >over an intuitionistic set theory at the cost of double negation. >However, in ASD, where open and closed subspaces are related via >continuous functions, and not set-theoretic complementation, the >Phoa principle allows this switch to be made without the not-not. > >In the case of a compact overt space such as the interval [0,1], >the classical, constructive, compact and over definitions of >connectedness agree. > >For a space that is either not compact or not overt, one of the >hypotheses must remain as an equation on the left of |-. > >Then constructive and overt connectedness agree: > U cup V = X |- <>U and <>V implies <>(U cap V) > >Compact connectedness is > U cap V = 0 |- [](U cup V) implies []U or []V. > >The latter gives rise to another approximate intermediate value >theorem: if f:K->R takes values >=0 and <=0 on OCCUPIED >subspaces, then its space of zeroes is also occupied. > >Here, OCCUPIED is the name that I propose for compact spaces whose >terminal projection is a proper surjection, just as an INHABITED >space is an overt one with an open surjection to 1. An occupied >space need not have any points. > >So far, I have only mentioned BINARY notions of connectedness, >but if we want to talk about families of connected COMPONENTS >then we must also consider INFINITARY connectedness (as Marta >stressed). Here the results for the constructive real line are >somewhat surprising. > >In order to avoid dependent types, I have found it more convenient >to discuss infinitary connectedness in terms of equivalence relations. > >In classical analysis, any OPEN EQUIVALENCE RELATION on [0,1], >ie any open subspace of the square that includes the diagonal, >is symmetric in it and has the transitivity property, is >INDISCRIMINATE - it relates 0 to 1 and indeed any point to any >other. > >This is not the case in Bishop's or Russian Recursive Analysis. >There is an open equivalence relation on [0,1] with infinitely >many equivalence classes, ie the interval fails the infinite >connectedness condition. The quotient by this equivalence >relation, ie the space that indexes the components, is discrete >but not Hausdorff, ie it admits an equality relation that is >not decidable. > >This one of the reasons why, at variance with many constructive >analysts, I believe that the HEINE--BOREL theorem is a necessary >part of analysis. > >In ASD, which obeys Heine--Borel, any open equivalence relation >on [0,1] or R is indiscriminate, as in the classical situation, >and the line and interval are connected in the infinitary senses. >Moreover, any open subspace of R is the disjoint union of countably >many open intervals, where each of these words needs careful >constructive re-definition. > >These results are in my paper "A lambda calculus for real analysis", >which was presented at CCA 2005 and you can obtain from > www.PaulTaylor.EU/ASD >I should point out that I am at the moment re-writing part of this >paper, to include a "need to know" introduction to continuous lattices, >cf my recent posting on this. However, the results that I have >discussed above are in the "stable" part of the text. > >Paul Taylor > > > _________________________________________________________________ Express yourself with free Messenger emoticons. Check out freemessengeremoticons.ca From rrosebru@mta.ca Mon Oct 15 10:12:37 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 15 Oct 2007 10:12:37 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IhPiS-0007dd-O8 for categories-list@mta.ca; Mon, 15 Oct 2007 10:10:41 -0300 Content-Disposition: inline Content-Transfer-Encoding: quoted-printable Content-Type: text/plain; charset="us-ascii" MIME-Version: 1.0 From: To: categories@mta.ca Subject: categories: Re: connectedness Date: Sun, 14 Oct 2007 19:56:42 -0400 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 73 Paul's remarks are quite cogent. Indeed, when Steve Schanuel and I introduced the notion of Extensive category, one of the main motivations was the recognition that a rational theory of connectedness requires a condition on the category of not-necessarily connected things, and moreover that a category like (K-rigs)^op satisfies this=20 condition even though it is not exact. (Also there=20 was the realization of a need for an algebraic=20 geometry for some cases where K is not a ring, indeed where it may satisfy 1+1=3D1). When C^op is an algebraic theory, i.e., C has=20 finite coproducts, then the algebras form a topos iff=20 those coproducts satisfy extensivity (because=20 then the attempt to consider "finite disjoint covers"=20 actually succeeds to satisfy Grothendieck's condition for a"topology"). But a non-trivial dual question is "almost" stated by Paul: For which algebraic categories is the opposite extensive ? Obvious extensions of K-rigs are M-K-rigs, where the=20 given monoid M acts by K-rig homomorphism, and an infinitesimal version of that where M is a Lie algebra acting by derivations. A special case of the question is, given an algebraic category that is coextensive, which varieties in it are also ? (Here I take "variety" in the original Birkhoff spirit, i.e., a full subcategory=20 that is also algebraic for the special reason that it is defined by a quotient theory and is thus closed wrt subalgebras, which a general full reflective algebraic subcategory would not be). A sufficient condition is that the inclusion functor is also COREFLECTIVE. Call these "core varieties". Proposition : A core variety in a coextensive algebraic category is also coextensive in its own right. Hence any core variety is a candidate to serve as the algebras for an algebraic geometry. (Extensivity was the only distinctive feature of rings mentioned in Gaeta's notes on Grothendieck's Buffalo Lectures 1973, and indeed you can verify that the basic=20 construction of a corresponding topos of spaces works, in=20 particular that the algebras become algebras of functions on these). For single-sorted theories, a core variety is defined by the=20 imposition of further identities in one variable having the rare property that the elements satisfying them form a subalgebra. For example ( )^p=3Did in algebras of characteristic p. Already for K=3D2, there are nontrivial core varieties in K-rigs. The best known is the category of distributive lattices, the corresponding topos of spaces being generated by the category of finite posets. The core of any 2-rig is the DL defined by two equations,=20 one of which is idempotence of the multiplication. But the other equation, taken alone, defines a larger core variety whose spaces look like intervals, cubes,etc ; it is intimately related to a less systematic subject burdened with the odd name "tropical". Bill On Fri Oct 12 9:53 , Paul Taylor sent: >Vaughan Pratt's original enquiry was actually in the context of >graph theory (as I suspected at the time, and he subsequently >confirmed), but I would like to add something from the point of >view of constructive real analysis. > >First, though, I would like to underline something that Steve Lack >(almost) said, namely that the category in which you index your >components, and therefore also the one in which you define >connectedness, need to be EXTENSIVE, ie their coproducts should >be disjoint, and stable under pullback, and the initial object strict. > >Maybe we've over-done philology recently, but "component" means >"putting together", where we expect the parts to cover the whole >(coproduct), without overlapping (disjoint), to be distinguishable >(like disjoint union, but unlike addition and disjunction). >The modern notion of extensivity, in which Steve had a part, >captures this idea very neatly. The equivalence between definitions >of connectedness based on 1+1 and on X+Y surely depends on stability >under pullback, and the requirement that the choice between left >and right be unique surely requires disjointness. Maybe a close >study of Marta Bunga's work on abstract connectedness would clarify >this. > >Vaughan originally asked about various categories of algebras, >and Steve mentioned commutative rings, but quietly turned their >arrows around. Stone duality would suggest to me that one should >look for connectedness of algebras in their OPPOSITE category of >"spaces", which I understand in a generic sense that includes >sets, graphs, predomains, locales and affive varieties. > ... From rrosebru@mta.ca Mon Oct 15 10:13:17 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 15 Oct 2007 10:13:17 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IhPkr-0000SF-7V for categories-list@mta.ca; Mon, 15 Oct 2007 10:13:09 -0300 Date: Sun, 14 Oct 2007 20:07:55 -0700 From: John Baez To: categories Subject: categories: week257 Message-ID: <20071015030755.GA8078@math.ucr.edu> Mime-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Content-Disposition: inline User-Agent: Mutt/1.4.2.1i Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 74 Dear Categorists - The latest issue of This Week's Finds contains enough category theory that I felt like sharing it here, especially since it contains links to the Catsters videos and my seminar with Jim Dolan on geometric representation theory. This seminar will ultimately discuss=20 "groupoidification", an idea also discussed in This Week's Finds starting in "week247". The web version has some pretty pictures. Best, jb ................................................................... Also available as http://math.ucr.edu/home/baez/week257.html October 14, 2007 This Week's Finds in Mathematical Physics (Week 257) John Baez Time flies! This week I'll finally finish saying what I did on=20 my summer vacation. After my trip to Oslo I stayed in London,=20 or more precisely Greenwich. While there, I talked with some good=20 mathematicians and physicists: in particular, Minhyong Kim, Ray=20 Streater, Andreas Doering and Chris Isham. I also went to a=20 topology conference in Sheffield... and Eugenia Cheng explained some cool stuff on the train ride there. I want to tell you about=20 all this before I forget. Also, the Tale of Groupoidification has taken a shocking new turn: it's now becoming available as a series of *videos*. But first, some miscellaneous fun stuff on math and astronomy. =20 Math: if you haven't seen a sphere turn inside out, you've got=20 to watch this classic movie, now available for free online: 1) The Geometry Center, Outside in, http://video.google.com/videoplay?docid=3D-6626464599825291409 Astronomy: did you ever wonder where dust comes from? I'm=20 not talking about dust bunnies under your bed - I'm talking=20 about the dust cluttering our galaxy, which eventually clumps=20 together to form planets and... you and me! These days most dust comes from aging stars called "asymptotic giant branch" stars. The sun will eventually become one of these. The story goes like this: first it'll keep burning until the hydrogen in its core is exhausted. Then it'll cool and become a red giant. Eventually helium at the core will ignite, and the Sun will shrink=20 and heat up again... but its core will then become cluttered with even=20 heavier elements, so it'll cool and expand once more, moving onto the "asymptotic giant branch". At this point it'll have a layered structure: heavier elements near the bottom, then a layer of helium, then hydrogen on the top. (A similar fate awaits any star between 0.6 and 10 solar masses, though the details depend on the mass. For the more dramatic fate of heavier stars, see "week204".) This layered structure is unstable, so asymptotic giant branch=20 stars pulse every 10 to 100 thousand years or so. And, they=20 puff out dust! Stellar wind then blows this dust out into space. =20 A great example is the Red Rectangle: 2) Rungs of the Red Rectangle, Astronomy picture of the day,=20 May 13, 2004, http://apod.nasa.gov/apod/ap040513.html Here two stars 2300 light years from us are spinning around each other while pumping out a huge torus of icy dust grains and hydrocarbon molecules. It's not really shaped like a rectangle=20 or X - it just looks that way. The scene is about 1/3 of a light=20 year across. Ciska Markwick-Kemper is an expert on dust. She's an astrophysicist at the University of Manchester. Together with some coauthors, she wrote a paper about the Red Rectangle: 3) F. Markwick-Kemper, J. D. Green, E. Peeters, Spitzer=20 detections of new dust components in the outflow of the Red=20 Rectangle, Astrophys. J. 628 (2005) L119-L122. Also available as arXiv:astro-ph/0506473. They used the Spitzer Space Telescope - an infrared telescope on=20 a satellite in earth orbit - to find evidence of magnesium and=20 iron oxides in this dust cloud. =20 But, what made dust in the early Universe? It took about a billion years after the Big Bang for asymptotic giant branch stars to form. But we know there was a lot of dust even before then! We can see it in distant galaxies lit up by enormous black holes=20 called "quasars", which pump out vast amounts of radiation as=20 stuff falls into them. =20 Markwick-Kemper and coauthors have also tackled that question: 4) F. Markwick-Kemper, S. C. Gallagher, D. C. Hines and J. Bouwman,=20 Dust in the wind: crystalline silicates, corundum and periclase in=20 PG 2112+059, Astrophys. J. 668 (2007), L107-L110. Also available as arXiv:0710.2225. They used spectroscopy to identify various kinds of dust in=20 a distant galaxy: a magnesium silicate that geologists call=20 "forsterite", a magnesium oxide called "periclase", and aluminum oxide, otherwise known as "corundum" - you may have seen it on=20 sandpaper. And, they hypothesize that these dust grains were formed in the hot wind emanating from the quasar at this galaxy's core! So, besides being made of star dust, as in the Joni Mitchell song, you also may contain a bit of black hole dust.=20 Okay - now that we've got that settled, on to London! Minhyong Kim is a friend I met back in 1986 when he was a grad=20 student at Yale. After dabbling in conformal field theory, he became a student of Serge Lang and went into number theory. He=20 recently moved to England and started teaching at University=20 College, London. I met him there this summer, in front of the=20 philosopher Jeremy Bentham, who had himself mummified and stuck in a wooden cabinet near the school's entrance. If you're not into number theory, maybe you should read this: 5) Minhyong Kim, Why everyone should know number theory, available at http://www.ucl.ac.uk/~ucahmki/numbers.pdf Personally I never liked the subject until I realized it was a form of *geometry*. For example, when we take an equation like this x^2 + y^3 =3D 1 and look at the real solutions, we get a curve in the plane -=20 a "real curve". If we look at the complex solutions, we get something bigger. People call it a "complex curve", because=20 it's analogous to a real curve. But topologically, it's=20 2-dimensional. This will be important in a few minutes, so=20 don't forget it! If we use polynomial equations with more variables, we get=20 higher-dimensional shapes called "algebraic varieties" - either=20 real or complex. Either way, we can study these shapes using=20 geometry and topology. =20 But in number theory, we might study the solutions of these=20 equations in some other number system - for example in Z/p,=20 meaning the integers modulo some prime p. At first glance there's=20 no geometry involved anymore. After all, there's just a *finite=20 set* of solutions! However, algebraic geometers have figured=20 out how to apply ideas from geometry and topology, mimicking=20 tricks that work for the real and complex numbers. =20 All this is very fun and mind-blowing - especially when we reach Grothendieck's idea of "etale topology", developed around 1958. =20 This is a way of studying "holes" in things like algebraic=20 varieties over finite fields. Amazingly, it gives results that=20 nicely match the results we get for the corresponding complex algebraic varieties! That's part of what the "Weil conjectures" say. You can learn the details here: 6) J. S. Milne, Lectures on Etale Cohomology, available at http://www.jmilne.org/math/CourseNotes/math732.html Anyway, I quizzed about Minhyong about one of the big mysteries that's been puzzling me lately. I want to know why the integers=20 resemble a 3-dimensional space - and how prime numbers are like=20 "knots" in this space! =20 Let me try to explain this in a very sketchy way, without getting=20 into any technical details. I'll still make mistakes... but this=20 stuff is just too cool to keep secret - so if the experts don't=20 explain it, nonexperts like me have to try. You can think of Z/p as giving a very simple sort of curve. =20 Naively you could imagine it as shaped like a ring, for example=20 the integers mod 7 here: 0 6 1 =20 5 2 4 3 =20 But now it's better to think of Z/p as a "line". After=20 all, a line is defined by one variable and no equations. Here=20 we have one variable in Z/p. =20 But remember: a curve defined in a field like Z/p acts a lot=20 like a complex curve. And, a complex curve is topologically=20 2-dimensional! =20 So, the "line" associated to Z/p seems 2-dimensional from the=20 viewpoint of etale topology. In other words, it's really more=20 like a "plane" - just like the complex numbers are topologically a=20 plane. This is true for each prime p. But the integers, Z, are more=20 complicated than any of these Z/p's. To be precise, we have maps Z -> Z/p for each p. So, if we think of Z as a kind of space, it's a big=20 space that contains all the "planes" corresponding to the Z/p's. =20 So, it's 3-dimensonal! =20 In short: from the viewpoint of etale topology, the integers have=20 one dimension that says which prime you're at, and two more coming=20 from the plane-like nature of each individual Z/p. =20 Naively you might imagine a stack of planes, one for each prime. =20 But that's a very crude picture, and it misses a crucial fact: the=20 primes get "tangled up" with each other. In fact, each "plane" has=20 a specially nice circle in it, and these circles are *linked*. =20 I've been fascinated by this ever since I heard about it, but I got even more interested when I saw a draft of a paper by=20 Kapranov and some coauthor. I got it from Thomas Riepe, who got it from Yuri Manin. I don't have it right here with me, so I'll add a reference later... but I don't think it's available yet, so the reference won't do you much good anyway. In this paper, the authors explain how the "Legendre symbol" of=20 primes is analogous to the "linking number" of knots. The Legendre symbol depends on two primes: it's 1 or -1 depending=20 on whether or not the first is a square modulo the second. The=20 linking number depends on two knots: it says how many times the=20 first winds around the second. The linking number stays the same when you switch the two knots. =20 The Legendre symbol has a subtler symmetry when you switch the=20 two primes: this symmetry is called "quadratic reciprocity", and=20 it has lots of proofs, starting with a bunch by Gauss - all a bit=20 tricky. =20 I'd feel very happy if I truly understood why quadratic reciprocity=20 reduces to the symmetry of the linking number when we think of=20 primes as analogous to knots. Unfortunately, I'll need to think a=20 lot more before I really get the idea. I got into number theory=20 late in life, so I'm pretty slow at it. =20 This paper studies subtler ways in which primes can be "linked": 7) Masanori Morishita, Milnor invariants and Massey products for=20 prime numbers, Compositio Mathematica 140 (2004), 69-83. You may know the Borromean rings, a design where no two rings are linked in isolation, but all three are when taken together. Here=20 the linking numbers are zero, but the linking can be detected by=20 something called the "Massey triple product". Morishita=20 generalizes this to primes! But I want to understand the basics... The secret 3-dimensional nature of the integers and certain other=20 "rings of algebraic integers" seems to go back at least to the work=20 of Artin and Verdier: 8) Michael Artin and Jean-Louis Verdier, Seminar on etale cohomology=20 of number fields, Woods Hole, 1964.=20 You can see it clearly here, starting in section 2: 9) Barry Mazur, Notes on the etale cohomology of number fields, Annales Scientifiques de l'Ecole Normale Superieure Ser. 4,=20 6 (1973), 521-552. Also available at http://www.numdam.org/numdam-bin/fitem?id=3DASENS_1973_4_6_4_521_0 By now, a big "dictionary" relating knots to primes has been=20 developed by Kapranov, Mazur, Morishita, and Reznikov. This=20 seems like a readable introduction: 10) Adam S. Sikora, Analogies between group actions on 3-manifolds and number fields, available as arXiv:math/0107210. I need to study it. These might also be good - I haven't looked at them yet: 11) Masanori Morishita, On certain analogies between knots and=20 primes, J. Reine Angew. Math. 550 (2002), 141-167. Masanori Morishita, On analogies between knots and primes,=20 Sugaku 58 (2006), 40-63. After giving a talk on 2-Hilbert spaces at University College, I went to dinner with Minhyong and some folks including Ray Streater. Ray Streater and Arthur Wightman wrote the book "PCT, Spin, Statistics and All That". Like almost every mathematician who has seriously tried to understand quantum field theory, I've learned a lot from this book. So, it was fun meeting Streater, talking with him - and finding out he'd once been made an honorary colonel of the US Army to get a free plane trip to the Rochester Conference! This was a big important particle physics conference, back in the good old days. He also described Geoffrey Chew's Rochester conference talk on the=20 analytic S-matrix, given at the height of the bootstrap theory fad.=20 Wightman asked Chew: why assume from the start that the S-matrix was=20 analytic? Why not try to derive it from simpler principles? Chew=20 replied that "everything in physics is smooth". Wightman asked about smooth functions that aren't analytic. Chew thought a moment and=20 replied that there weren't any. Ha-ha-ha... What's the joke? Well, first of all, Wightman had already succeeded in deriving the analyticity of the S-matrix from simpler principles.=20 Second, any good mathematician - but not necessarily every physicist,=20 like Chew - will know examples of smooth functions that aren't=20 analytic.=20 Anyway, Streater has just finished an interesting book on "lost=20 causes" in physics: ideas that sounded good, but never panned out. =20 Of course it's hard to know when a cause is truly lost. But a=20 good pragmatic definition of a lost cause in physics is a topic=20 that shouldn't be given as a thesis problem. =20 So, if you're a physics grad student and some professor wants you to=20 work on hidden variable theories, or octonionic quantum mechanics,=20 or deriving laws of physics from Fisher information, you'd better=20 read this: 11) Ray F. Streater, Lost Causes in and Beyond Physics, Springer=20 Verlag, Berlin, 2007. (I like octonions - but I agree with Streater about not inflicting=20 them on physics grad students! Even though all my students are in=20 the math department, I still wouldn't want them working mainly on=20 something like that. There's a lot of more general, clearly useful=20 stuff that students should learn.)=20 I also spoke to Andreas Doering and Chris Isham about their work=20 on topos theory and quantum physics. Andreas Doering lives near Greenwich, while Isham lives across the Thames in London proper. So, I talked to Doering a couple times, and once we visited Isham at his house. I mainly mention this because Isham is one of the gurus of quantum gravity, profoundly interested in philosophy... so I was surprised, at the end of our talk, when he showed me into a room with a huge=20 rack of computers hooked up to a bank of about 8 video monitors, and controls reminiscent of an airplane cockpit. It turned out to be his homemade flight simulator! He's been a=20 hobbyist electrical engineer for years - the kind of guy who=20 loves nothing more than a soldering iron in his hand. He'd just=20 gotten a big 750-watt power supply, since he'd blown out his previous one. =20 Anyway, he and Doering have just come out with a series of papers: 11) Andreas Doering and Christopher Isham, A topos foundation=20 for theories of physics: I. Formal languages for physics,=20 available as arXiv:quant-ph/0703060. II. Daseinisation and the liberation of quantum theory,=20 available as arXiv:quant-ph/0703062. III. The representation of physical quantities with arrows, available as arXiv:quant-ph/0703064. IV. Categories of systems, available as arXiv:quant-ph/0703066. Though they probably don't think of it this way, you can think=20 of their work as making precise Bohr's ideas on seeing the quantum world through classical eyes. Instead of talking about all observables at once, they consider collections of observables that you can measure simultaneously without the uncertainty principle kicking in. These collections are called "commutative subalgebras".=20 You can think of a commutative subalgebra as a classical snapshot of the full quantum reality. Each snapshot only shows part of the reality. One might show an electron's position; another might show its momentum. Some commutative subalgebras contain others, just like some open=20 sets of a topological space contain others. The analogy is a good=20 one, except there's no one commutative subalgebra that contains *all* the others. =20 Topos theory is a kind of "local" version of logic, but where the=20 concept of locality goes way beyond the ordinary notion from=20 topology. In topology, we say a property makes sense "locally"=20 if it makes sense for points in some particular open set. In the Doering-Isham setup, a property makes sense "locally" if it makes sense "within a particular classical snapshot of reality" - that is, relative to a particular commutative subalgebra. (Speaking of topology and its generalizations, this work on topoi and=20 physics is related to the "etale topology" idea I mentioned a while=20 back - but technically it's much simpler. The etale topology lets you define a topos of "sheaves" on a certain category. The=20 Doering-Isham work just uses the topos of "presheaves" on the poset of commutative subalgebras. Trust me - while this may sound scary,=20 it's much easier.) =20 Doering and Isham set up a whole program for doing physics=20 "within a topos", based on existing ideas on how to do math in=20 a topos. You can do vast amounts of math inside any topos just=20 as if you were in the ordinary world of set theory - but using=20 intuitionistic logic instead of classical logic. Intuitionistic logic denies the principle of excluded middle, namely: "For any statement P, either P is true or not(P) is true." In Doering and Isham's setup, if you pick a commutative subalgebra=20 that contains the position of an electron as one of its observables, it can't contain the electron's momentum. That's because these observables don't commute: you can't measure them both simultaneously. So, working "locally" - that is, relative to this particular=20 subalgebra - the statement P =3D "the momentum of the electron is zero" is neither true nor false! It's just not defined. Their work has inspired this very nice paper: 12) Chris Heunen and Bas Spitters, A topos for algebraic quantum theory, available as arXiv:0709.4364. so let me explain that too. I said you can do a lot of math inside a topos. In particular,=20 you can define an algebra of observables - or technically, a "C*-algebra". By the Isham-Doering work I just sketched, any C*-algebra of=20 observables gives a topos. Heunen and Spitters show that=20 the original C*-algebra gives rise to a commutative C*-algebra in this topos, even if the original one was=20 noncommutative! That actually makes sense, since in this setup, each "local view"=20 of the full quantum reality is classical. What's really neat is=20 that the Gelfand-Naimark theorem, saying commutative C*-algebras=20 are always algebras of continuous functions on compact Hausdorff=20 spaces, can be generalized to work within any topos. So, we get=20 a space *in our topos* such that observables of the C*-algebra=20 *in the topos* are just functions on this space. =20 I know this sounds technical if you're not into this stuff. But it's really quite wonderful. It basically means this: using topos=20 logic, we can talk about a classical space of states for a quantum=20 system! However, this space typically has "no global points". In=20 other words, there's no overall classical reality that matches all=20 the classical snapshots. =20 As you can probably tell, category theory is gradually seeping into this post, though I've been doing my best to keep it hidden. Now I want to say what Eugenia Cheng explained on=20 that train to Sheffield. But at this point, I'll break down and assume you know some category theory - for example, monads. If you don't know about monads, never fear! I defined them in=20 "week89", and studied them using string diagrams in "week92".=20 Even better, Eugenia Cheng and Simon Willerton have formed a=20 little group called the Catsters - and under this name, they've=20 put some videos about monads and string diagrams onto YouTube! =20 This is a really great new use of technology. So, you should=20 also watch these: 14) The Catsters, Monads,=20 http://youtube.com/view_play_list?p=3D0E91279846EC843E The Catsters, Adjunctions,=20 http://youtube.com/view_play_list?p=3D54B49729E5102248 The Catsters, String diagrams, monads and adjunctions, http://youtube.com/view_play_list?p=3D50ABC4792BD0A086 A very famous monad is the "free abelian group" monad F: Set -> Set which eats any set X and spits out the free abelian group on X,=20 say F(X). A guy in F(X) is just a formal linear combination of guys in X, with integer coefficients. Another famous monad is the "free monoid" monad=20 G: Set -> Set This eats any set X and spits out the free monoid on X, namely=20 G(X). A guy in G(X) is just a formal product of guys in X. Now, there's yet another famous monad, called the "free=20 ring" monad, which eats any set X and spits out the free ring on this set. But, it's easy to see that this is just F(G(X))! After all, F(G(X)) consists of formal linear combinations of formal products of guys in X. But that's precisely what you find in the free ring on X. =20 But why is FG a monad? There's more to a monad than just a=20 functor. A monad is really a kind of *monoid* in the world of functors from our category (here Set) to itself. In particular,=20 since F is a monad, it comes with a natural transformation called a "multiplication": m: FF =3D> F which sends formal linear combinations of formal linear combinations to formal linear combinations, in the obvious way. Similarly, since G is a monad, it comes with a natural transformation n: GG =3D> G sending formal products of formal products to formal products. But how does FG get to be a monad? For this, we need some=20 natural transformation from FGFG to FG! There's an obvious thing to try, namely mn=20 FGFG =3D=3D=3D=3D=3D=3D> FFGG =3D=3D=3D=3D=3D=3D> FG where in the first step we switch G and F somehow, and in the second step we use m and n. But, how do we do the first step? We need a natural transformation d: GF =3D> FG which sends formal products of formal linear combinations to formal linear combinations of formal products. Such a thing obviously exists; for example, it sends (x + 2y)(x - 3z)=20 to xx + 2yx - 3xz - 6yz It's just the distributive law! =20 Quite generally, to make the composite of monads F and G=20 into a new monad FG, we need something that people call a "distributive law", which is a natural transformation d: GF =3D> FG This must satisfy some equations - but you can work out those yourself. For example, you can demand that FdG mn=20 FGFG =3D=3D=3D=3D=3D=3D> FFGG =3D=3D=3D=3D=3D=3D> FG make FG into a monad, and see what that requires. Besides the=20 "multiplication" in our monad, we also need the "unit", so you=20 should also think about that - I'm ignoring it here because it's less sexy than the multiplication, but it's equally essential. However: all this becomes more fun with string diagrams! As the Catsters explain, and I explained in "week89", the=20 multiplication m: FF =3D> F can be drawn like this: \ / \ / F\ F/ \ / \ / \ / \ / \ / |m =20 | | | | | F| | And, it has to satisfy the associative law, which says we get the same answer either way when we multiply three things: \ / / \ \ / \ / / \ \ / F\ /F F/ F\ F\ /F \/ / \ \/ m\ / \ /m=20 \ / \ / F\ / \ /F \ / \ / |m |m | | | =3D | | | | | | | F| F| | | The multiplication n: GG =3D> G looks similar to m, and it too has to satisfy the associative law. =20 How do we draw the distributive law d: FG =3D> GF? Since it's a=20 process of switching two things, we draw it as a *braiding*: F\ /G \ / /=20 / \ G/ \F=20 I hope you see how incredibly cool this is: the good old=20 distributive law is now a *braiding*, which pushes our diagrams into the third dimension! =20 Given this, let's draw the multiplication for our would-be monad FG, namely=20 FdG mn=20 FGFG =3D=3D=3D=3D=3D=3D> FFGG =3D=3D=3D=3D=3D=3D> FG It looks like this: \ \ / / \ \ / / F\ G\ F/ /G \ \ / / \ \ / / \ \ / / \ / / \ / \ / |m |n =20 | | | | | | | | | | F| |G | | Now, we want *this* multiplication to be associative! So,=20 we need to draw an equation like this: \ / / \ \ / \ / / \ \ / \ / / \ \ / \/ / \ \/ \ / \ /=20 \ / \ / \ / \ / \ / \ / | | | | | =3D | | | | | | | | | | |=20 but with the strands *doubled*, as above - I'm too lazy to draw=20 this here. And then we need to find some nice conditions that=20 make this associative law true. Clearly we should use the=20 associative laws for m and n, but the "braiding" - the=20 distributive law d: FG =3D> GF - also gets into the act. I'll leave this as a pleasant exercise in string diagram=20 manipulation. If you get stuck, you can peek in the back of=20 the book: 14) Wikipedia, Distibutive law between monads,=20 http://en.wikipedia.org/wiki/Distributive_law_between_monads The two scary commutative rectangles on this page are the=20 "nice conditions" you need. They look nicer as string=20 diagrams. One looks like this: F\ G\ /G F\ G/ /G \ \ / \ / / \ |n \ / / \ / / / \ / =3D / \ / / / / / \ / /\=20 / \ \ / \ / \ \ / \ G/ \F |n \F / \ G| \ In words:=20 "multiply two G's and slide the result over an F" =3D "slide both the G's over the F and then multiply them" If the pictures were made of actual string, this would be obvious! The other condition is very similar. I'm too lazy to draw it, but it says=20 "multiply two F's and slide the result under a G" =3D=20 "slide both the F's under a G and then multiply them" All this is very nice, and it goes back to a paper by Beck: 15) Jon Beck, Distributive laws, Lecture Notes in Mathematics=20 80, Springer, Berlin, pp. 119=96140.=20 This isn't what Eugenia explained to me, though - I already knew this stuff. She started out by explaining something in a paper=20 by Street: 16) Ross Street, The formal theory of monads, J. Pure Appl. Alg. 2 (1972), 149-168. which is reviewed at the beginning here: 17) Steve Lack and Ross Street, The formal theory of monads II, J. Pure Appl. Alg. 175 (2002), 243-265. Also available at http://www.maths.usyd.edu.au/u/stevel/papers/ftm2.html (Check out the cool string diagrams near the end!) =20 Street noted that for any category C, there's a category Mnd(C)=20 whose objects are monads on C and whose morphisms are "monad transforms": functors from C to C that make an obvious square commute. =20 And, he noted that a monad on Mnd(C) is a pair of monads on C related by a distributive law! That's already mindbogglingly beautiful. According to Eugenia, it's in the last sentence of Street's paper. But in her new work: 18) Eugenia Cheng, Iterated distributive laws, available as arXiv:0710.1120. she goes a bit further: she considers monads in Mnd(Mnd(C)),=20 and so on. Here's the punchline, at least for today: she shows=20 that a monad in Mnd(Mnd(C)) is a triple of monads F, G, H related=20 by distributive laws satisfying the Yang-Baxter equation: \F G/ |H F| G\ /H \ / | | \ / / | | / / \ | | / \ / \ | \ / \ | \ / \ / | | / =3D / |=20 | / \ / \ | | / \ / \ | \ / | | \ / \ / | | \ / / | | / / \ | | / \ /H \G |F H| G/ \F This is also just what you need to make the composite FGH into a monad! By the way, the pathetic piece of ASCII art above is lifted=20 from "week1", where I first explained the Yang-Baxter equation. That was back in 1993. So, it's only taken me 14 years to learn that you can derive this equation from considering monads on the category of monads on the category of monads on a category. You may wonder if this counts as progress - but Eugenia studies lots of *examples* of this sort of thing, so it's far from pointless. =20 Okay... finally, the Tale of Groupoidification. I'm a bit tired now, so instead of telling you more of the tale, let me just say the big news. Starting this fall, James Dolan and I are running a seminar on geometric representation theory, which will discuss: Actions and representations of groups, especially symmetric groups Hecke algebras and Hecke operators Young diagrams Schubert cells for flag varieties q-deformation=20 Spans of groupoids and groupoidification This is the Tale of Groupoidification in another guise. Moreover, the Catsters have inspired me to make videos of this=20 seminar! You can already find some here, along with course=20 notes and blog entries where you can ask questions and talk about=20 the material: 19) John Baez and James Dolan, Geometric representation theory seminar, http://math.ucr.edu/home/baez/qg-fall2007/ More will show up in due course. I hope you join the fun. ----------------------------------------------------------------------- Quote of the Week: It is a glorious feeling to discover the unity of a set of phenomena that at first seem completely separate. - Albert Einstein ----------------------------------------------------------------------- Previous issues of "This Week's Finds" and other expository articles on mathematics and physics, as well as some of my research papers, can be obtained at http://math.ucr.edu/home/baez/ For a table of contents of all the issues of This Week's Finds, try http://math.ucr.edu/home/baez/twfcontents.html A simple jumping-off point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html If you just want the latest issue, go to =20 http://math.ucr.edu/home/baez/this.week.html From rrosebru@mta.ca Mon Oct 15 21:44:10 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 15 Oct 2007 21:44:10 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IhaLb-0005zj-Qc for categories-list@mta.ca; Mon, 15 Oct 2007 21:31:47 -0300 Date: Mon, 15 Oct 2007 17:07:53 -0700 From: John Baez To: categories Subject: categories: week257 Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline User-Agent: Mutt/1.4.2.1i Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 75 Dear Categorists - I made some mistakes in my account of Cheng's work, saying "monad on a category" at some points where I should have said "monad in a 2-category". Here's a fixed version: Street noted that we can talk about monads, not just in the 2-category of categories, but in any 2-category. I actually explained monads at this level of generality back in "week89". Indeed, for any 2-category C, there's a 2-category Mnd(C) of monads in C. And, he noted that a monad in Mnd(C) is a pair of monads in C related by a distributive law! That's already mindbogglingly beautiful. According to Eugenia, it's practically the last sentence of Street's paper. But in her new work: 18) Eugenia Cheng, Iterated distributive laws, available as arXiv:0710.1120. she goes a bit further: she considers monads in Mnd(Mnd(C)), and so on. Here's the punchline, at least for today: she shows that a monad in Mnd(Mnd(C)) is a triple of monads F, G, H related by distributive laws satisfying the Yang-Baxter equation: \F G/ |H F| G\ /H \ / | | \ / / | | / / \ | | / \ / \ | \ / \ | \ / \ / | | / = / | | / \ / \ | | / \ / \ | \ / | | \ / \ / | | \ / / | | / / \ | | / \ /H \G |F H| G/ \F This is also just what you need to make the composite FGH into a monad! By the way, the pathetic piece of ASCII art above is lifted from "week1", where I first explained the Yang-Baxter equation. That was back in 1993. So, it's only taken me 14 years to learn that you can derive this equation from considering monads in the category of monads in the category of monads in a 2-category. Also, I should have given a reference to earlier work on Gelfand duality in a topos: Bernhard Banachewski and Christopher J. Mulvey, A globalisation of the Gelfand duality theorem, Ann. Pure Appl. Logic 137 (2006), 62-103. Also available at http://www.maths.sussex.ac.uk/Staff/CJM/research/pdf/globgelf.pdf They show that any commutative C*-algebra A in a Grothendieck topos is canonically isomorphic to the C*-algebra of continuous complex functions on the compact, completely regular locale that is its maximal spectrum (that is, the space of homomorphisms f: A -> C). Conversely, they show any compact completely regular locale X gives a commutative C*-algebra consisting of continuous complex functions on X. From rrosebru@mta.ca Mon Oct 15 21:44:11 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 15 Oct 2007 21:44:11 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IhaKX-0005tE-Nm for categories-list@mta.ca; Mon, 15 Oct 2007 21:30:41 -0300 From: "Marta Bunge" To: categories@mta.ca Subject: categories: RE: connectedness Date: Mon, 15 Oct 2007 09:29:32 -0400 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 76 Dear Paul, >First, though, I would like to underline something that Steve Lack >(almost) said, namely that the category in which you index your >components, and therefore also the one in which you define >connectedness, need to be EXTENSIVE, ie their coproducts should >be disjoint, and stable under pullback, and the initial object strict. > >Maybe we've over-done philology recently, but "component" means >"putting together", where we expect the parts to cover the whole >(coproduct), without overlapping (disjoint), to be distinguishable >(like disjoint union, but unlike addition and disjunction). >The modern notion of extensivity, in which Steve had a part, >captures this idea very neatly. The equivalence between definitions >of connectedness based on 1+1 and on X+Y surely depends on stability >under pullback, and the requirement that the choice between left >and right be unique surely requires disjointness. Maybe a close >study of Marta Bunge's work on abstract connectedness would clarify >this. > In my now obsolete 1966 thesis, the context was that of a category with finite limits and finite coproducts, but I had not assumed therein that coproducts should be disjoint and universal. With these assumptions, the definitions of `abstractly unary' for arbitrary binary products (factors through AT LEAST one of the injections) and of `abstractly exclusively unary' (factors through exactly one of the injections) are equivalent by the disjointness part, and are equivalent also to the same notions with binary coproducts of 1 instead of arbitrary binary coproducts (by the universal or stability part). So, *any* of those in this context should mean `connected'. Without those conditions, but with just a terminal object and binary coproducts, then the `at least' part does not reduce to that of coproducts of 1, but the `at most' part does. In that case, the notions correspond to `abstractly unary' and `abstractly exclusively unary', and they are not equivalent. So it all depends on the ambient category. > >So far, I have only mentioned BINARY notions of connectedness, >but if we want to talk about families of connected COMPONENTS >then we must also consider INFINITARY connectedness (as Marta >stressed). Here the results for the constructive real line are >somewhat surprising. > > I still have to digest your disquisitions on constructive analysis, which seem most interesting, but on the above point, I emphasize that inded one must keep the disctinction between connectedness wirt binary coproducts and connectedness wrt arbitrary coproducts (indexed externally, e.g. by a set in a Grothendieck topos E-->SET, or by an object of S in the cae of a bounded topos E-->S. Whether the terminology must make that distinction I am not sure of, or maybe we could say `connected' for the binary case (which is also intrinsic), and `S-connected' for the case of S-indexed coproducts. Once again, under enough hypotheses as I szaid above and was also mentioned by Steve Lack. I am not sure of which hypotheses Vaughan wants to make but, if the minimal possible, then he might need the detailed analysis that I proposed and, in that case, reserve `connected' in the binary case for `abstractly exclusively unary', not simply `abstractly unary', and `connected' when only the binary coproduct considered is 1+1. But if his categories ae categories of graphs, I don't see his problem. With best regards, Marta From rrosebru@mta.ca Tue Oct 16 21:30:43 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 16 Oct 2007 21:30:43 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ihwgu-0004U6-Ip for categories-list@mta.ca; Tue, 16 Oct 2007 21:23:16 -0300 From: Thomas Streicher Subject: categories: connectedness fibrationally To: categories@mta.ca Date: Tue, 16 Oct 2007 12:59:36 +0200 (CEST) MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 77 Recently it has been discussed what is the appropriate notion of connecteness for a category \X relative to a category \B. The following appears as natural to me. Let \B be a category and P : \X -> \B be a fibration of categories with a terminal object and with internal sums. Then for every object I in \B there is an obvious functor \Delta_I : \B/I -> \X_I sending u : J -> I to \coprod_u 1_J. An object X \in \X_I is an I_indexed family of connected objects iff the functor \X_I(X,\Delta_I(-)) : (\B/I)^\op -> Set is represented by \id_I, i.e there exists eta_X : X \to 1_I such that for every cocartesian arrow \phi : 1_J -> \Delta_I(u) over u : J -> I and vertical arrow \alpha : X -> \Delta_I(u) there exists a unique arrow s : I -> J making the diagram X --------------------> 1_I | | | \alpha | 1_s | | V cocart. V \Delta_I(u) <---------------1_J commute (where the top arrow is vertical). In case I = 1 (where we write \Delta for \Delta_I) this means that for every f : X -> Delta(I) there is a unique i : 1 -> I with f = \Delta(i) \circ eta_X. Notice that in case \B has finite limits, P is a fibration of categories with finite limits and stable and disjoint sums the fibration P is equivalent to \Delta^* P_{\X_1}. This is an old result of Moens (1982) (see section 15 of www.mathematik.tu-darmstadt.de/~streicher/FIBR/FibLec.pdf.gz for an exposition). Thomas From rrosebru@mta.ca Tue Oct 16 21:30:43 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 16 Oct 2007 21:30:43 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ihwi0-0004bX-Ua for categories-list@mta.ca; Tue, 16 Oct 2007 21:24:25 -0300 Date: Tue, 16 Oct 2007 11:07:37 -0600 (MDT) Subject: categories: Benford's Law From: mjhealy@ece.unm.edu To: categories@mta.ca MIME-Version: 1.0 Content-Type: text/plain;charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 78 This message was stimulated by John Baez's week257 which, though interesting as usual, has one item of special interest to me at this time= . I haven't yet looked at Minhyong Kim's work, and I don't know how this fits in with number theory or categories, but a friend is encouraging me to go to the following conference on Benford's Law:=20 http://www.ece.unm.edu/benford . Does anybody on this list (including you, John) know of a connection between Benford's Law and any work in category theory? I would really like to hear about it if so. Thanks, Mike From rrosebru@mta.ca Tue Oct 16 21:30:43 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 16 Oct 2007 21:30:43 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IhwhM-0004XF-VP for categories-list@mta.ca; Tue, 16 Oct 2007 21:23:45 -0300 From: Juergen Koslowski Subject: categories: CT07, please help identifying more participants! To: categories@mta.ca (categories list) Date: Tue, 16 Oct 2007 13:56:02 +0200 (CEST) MIME-Version: 1.0 Content-Type: text/plain; charset=iso-8859-1 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 79 Dear participants of CT07, Thanks for poining out a number of names I didn't know, especially of=20 the participants from Spain. Still, there are some persons that have=20 not been identified so far: page 8, top center, with David Kruml middle left, with Joa~o Jose' Xarez middle right, with Dominic Verity (perhaps Marie Bjerrum?) bottom left, with Gonc,alo Gutierres page 9, middle enter, with Lurdes Sousa page b, top center and right middle left, center and right (with Alejandro Ribera) bottom center and right, with Jeff Egger group: ?0 between William Boshuk and Tom Leinster ?7 behind Claudio Hermida and Mata Bunge ?8 right of George Janelidze, below Renato Betti ?9 below Jonathan Cohen, left of Michel Hebert ?b behind Marco Grandis, left of Rober Seely ?c right of Simona Paoli, behind left of Graham White ?e in front of Peter Johnstone ?f behind Pedro Resende ?g right of Maria Manuel Clementino, in front of Tim van der Lind= en ?h between Anders Kock and Eduardo Dubuc ?i behind Sandra Mantovani, left of Michael Makkai ?l right of Micheal Hyland, in front of Thorsten Palm=20 Best regards, -- J=FCrgen --=20 Juergen Koslowski If I don't see you no more on this world ITI, TU Braunschweig I'll meet you on the next one koslowj@iti.cs.tu-bs.de and don't be late! http://www.iti.cs.tu-bs.de/~koslowj Jimi Hendrix (Voodoo Child, SR) From rrosebru@mta.ca Wed Oct 17 14:15:22 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 17 Oct 2007 14:15:22 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IiCGK-0000q1-MI for categories-list@mta.ca; Wed, 17 Oct 2007 14:00:53 -0300 From: Juergen Koslowski Subject: categories: CT07, thanks for your help! To: categories@mta.ca (categories list) Date: Wed, 17 Oct 2007 11:44:56 +0200 (CEST) MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 80 Dear participants of CT07, Thanks for your help! We are down to 3 unidentified persons: page b, top center, right of Peter LeFanu Lumisdaine middle center, left of Charles Crissman group: ?9 below Jonathan Cohen, left of Michel Hebert Best regards, -- J=FCrgen -- Juergen Koslowski If I don't see you no more on this world ITI, TU Braunschweig I'll meet you on the next one koslowj@iti.cs.tu-bs.de and don't be late! http://www.iti.cs.tu-bs.de/~koslowj Jimi Hendrix (Voodoo Child, SR) From rrosebru@mta.ca Wed Oct 17 14:15:22 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 17 Oct 2007 14:15:22 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IiCFm-0000k8-Ig for categories-list@mta.ca; Wed, 17 Oct 2007 14:00:18 -0300 Date: Tue, 16 Oct 2007 19:51:49 -0700 From: John Baez To: categories Subject: categories: Benford's law Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 81 Mike writes: >I haven't yet looked at Minhyong Kim's work, and I don't know how this >fits in with number theory or categories, but a friend is encouraging me >to go to the following conference on Benford's Law: >http://www.ece.unm.edu/benford . > >Does anybody on this list (including you, John) know of a connection >between Benford's Law and any work in category theory? I would really >like to hear about it if so. I don't know any interesting connection between this law and category theory or number theory. I didn't know it was called "Benford's law", but I knew the idea: if you take a table of widely spread numbers (say the gross national products of nations, or the incomes of Americans), often about log 2 ~ 30% will have 1 as their first digit, about log 3 - log 2 ~ 17% will have 2 as their first digit, and so on. It's easy to derive this law from the assumption that the data is distributed in an approximately scale-invariant way within a certain range. (That is, the percentage of numbers in your table between X and cX is about equal to the percentage between Y and cY, for c not too big, and X and Y within some large but finite range. Or: the logarithms of the numbers are approximately uniformly distributed over some interval.) So, the mystery of Benford's law reduces to the mystery of this fact: in practice, widely spread numbers are often distributed in an approximately scale-invariant way, within some range. (Perhaps some people find Benford's law mysterious because it's impossible for a probability distribution to be *perfectly* scale-invariant. But that's a red herring. It's enough to have approximate scale-invariance within some range, for example a couple powers of 10.) Why is approximate scale-invariance so common? People have written books on this! Here's a nice one: Manfred Schroeder, Chaos, Fractals, Power Laws, W. H. Freeman, 1992. Or, for starters: http://en.wikipedia.org/wiki/Power_law I would rather go to a conference on power laws than a conference on Benford's law, which seems like just a spinoff. Best, jb From rrosebru@mta.ca Wed Oct 17 14:15:22 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 17 Oct 2007 14:15:22 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IiCGu-0000v3-3n for categories-list@mta.ca; Wed, 17 Oct 2007 14:01:28 -0300 Date: Wed, 17 Oct 2007 09:08:17 -0300 From: "Robert J. MacG. Dawson" MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: Benford's Law Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 82 mjhealy@ece.unm.edu wrote: > Does anybody on this list (including you, John) know of a connection > between Benford's Law and any work in category theory? I would really > like to hear about it if so. I doubt if there is much of one. I suppose _some_ sort of connection might be made through the concept of "invariance" - Benford's law holds for distributions that are wide enough not to have a "natural scale". If you cannot give an approximate answer to "how big is a (river/piece of string/data file/bank deposit)?" then the distribution of first digits in (say) centimeters should [waving hands hard] be the same as that in furlongs or wavelengths of green light; and from that property Benford's law follows. On the other hand, humans are approximately of a height, to the point that the foot, hand, cubit, fathom, etc. can be used as rough units in their natural form. Thus there is a natural scale for human heights, and we are not surprised that almost all human heights in meters have a first digit 1 and very few do in inches. -Robert From rrosebru@mta.ca Wed Oct 17 19:39:40 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 17 Oct 2007 19:39:40 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IiHTa-0002Xk-4T for categories-list@mta.ca; Wed, 17 Oct 2007 19:34:54 -0300 Date: Wed, 17 Oct 2007 19:11:26 +0200 From: Joachim Kock Subject: categories: Advanced Course on Simplicial Methods in Higher Categories To: categories@mta.ca MIME-version: 1.0 Content-type: text/plain; charset=iso-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 83 This is the second announcement of the Advanced Course on Simplicial Methods in Higher Categories February 4 to 14, 2008 Centre de Recerca Matem=E0tica Bellaterra (Barcelona) an event within the CRM thematic year on Homotopy Theory and Higher Categories http://www.crm.cat/hocat/ The Advanced Course consists of three lecture series: -- Andr=E9 Joyal (UQAM, Montr=E9al) "The theory of quasi-categories and its applications" -- Ieke Moerdijk (Utrecht) "Dendroidal sets" -- Bertrand To=EBn (Toulouse) "Simplicial presheaves and derived geometries" In the preceding week (28/1--1/2), two preparatory mini-courses are planned: Myles Tierney: "Simplicial homotopy theory", and Mathieu Anel: "The functor-of-points approach to geometry, and=20 stacks". The CRM offers a limited number of grants covering accommodation for young researchers. The deadline for application is October 31, 2007. Otherwise the deadline for registration is December 14. For further information, see http://www.crm.cat/HigherCategories/ Carles Casacuberta and Joachim Kock From rrosebru@mta.ca Thu Oct 18 22:57:33 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 18 Oct 2007 22:57:33 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Iigww-0002yR-AS for categories-list@mta.ca; Thu, 18 Oct 2007 22:46:54 -0300 Date: Thu, 18 Oct 2007 12:32:22 -0300 (ADT) From: Bob Rosebrugh To: categories Subject: categories: [cmath] Homotopy Theory Postdoctoral Fellowship - University of Western Ontario, Department of Mathematics (fwd) MIME-Version: 1.0 Content-Type: TEXT/PLAIN; CHARSET=US-ASCII; FORMAT=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 84 [Note from moderator: this may be of interest to some list members.] ---------- Forwarded message ---------- Date: Thu, 18 Oct 2007 11:16:09 -0400 (EDT) From: Alan Kelm To: CMath E-Mail Distribution List Subject: [cmath] Homotopy Theory Postdoctoral Fellowship - University of Western Ontario, Department of Mathematics The University of Western Ontario Department of Mathematics Homotopy Theory Postdoctoral Fellowship The Department of Mathematics at the University of Western Ontario has one postdoctoral position available in areas related to homotopy theory, including applications to other fields. The position is for a one-year term beginning July 1, 2008, with a flexible start date and the possibility of an extension to a second and third year subject to budgetary considerations. The salary will be $40,000 CDN per year plus a tax free research fund of $1,500. The position will involve teaching two half courses per year, in addition to research. The successful candidate will work under the supervision of J.F. Jardine, and should have completed a Ph.D. in 2005 or later. Candidates are encouraged to apply electronically using www.mathjobs.org. Applications may instead be mailed to: Professor D. M. Riley, Chair Department of Mathematics University of Western Ontario London, Ontario N6A 5B7 Canada All applications should include a curriculum vitae, a research statement, and at least three letters of reference. At least one letter of reference should comment on the teaching abilities of the applicant. E-mail inquiries and submissions may be sent to math-pos@uwo.ca. Information about the department can be found at http://www.math.uwo.ca. J.F. Jardine's home page is http://www.math.uwo.ca/~jardine/cv/index.html. The deadline for applications is January 15, 2008. --------------------------------------------------------------------------- This employment position is among those listed in the Employment section of the CMS website: http://cms.math.ca/Employment From rrosebru@mta.ca Sun Oct 21 19:25:51 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 21 Oct 2007 19:25:51 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Ijj0l-0003vT-Td for categories-list@mta.ca; Sun, 21 Oct 2007 19:11:07 -0300 From: "George Janelidze" To: "\"Categories\"" Subject: categories: Max Kelly Conference in Cape Town: SECOND ANNOUNCEMENT Date: Sat, 20 Oct 2007 01:49:55 +0200 MIME-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 85 MAX KELLY CONFERENCE IN CAPE TOWN January 2007: SECOND ANNOUNCEMENT ORGANIZING COMMITTEE: Guillaume Br=FCmmer John Frith Partha Ghosh Christopher Gilmour James Gray Keith Hardie David Holgate George, Tamar, and Zurab Janelidze Zechariah Mushaandja Peter Ouwehand Ingrid Rewitzky SCIENTIFIC COMMITTEE: Martin Hyland George Janelidze Michael Johnson Peter Johnstone Stephen Lack Ross Street Walter Tholen Richard Wood DATES: 1. Registration and submission of Abstracts: before 1 December 2007. 2. Opening of the Conference: Monday, 21 January 2008. 3. The last "official" talks will be given on Saturday, 26 January 2008, however for those who will stay for the next week various informal sessio= ns will be organized. 4. There will also be Conference Dinner and Evening of Memories probably = on Friday 25 January. The precise dates will be given later in the third announcement. 5. Paper submission for the Proceedings Volume: before 1 April 2008 ACCOMMODATION: Since all of January is a peak time for tourism in Cape Town, it is urgen= t that you book the accommodation as soon as possible. It is easy to do it independently through Internet of course. However, the following addition= al information should be helpful: We recommend that participants book either 1. All Africa House http://www.cal.uct.ac.za/ . This is very nice and reasonable accommodation on the campus within 10 mins walking distance of the conference venue. They are reserving 20 rooms for us until the end o= f October. Please contact Callie at prezandt@protem.uct.ac.za (Web: www.cal.uct.ac.za ) to make a reservation and mention that you are a participant in the Max Kelley Conference. 2. Alternatively, if you prefer to stay in the City, we recommend the Victoria and Alfred Waterfront http://www.waterfront.co.za/ which has a number of hotels (some quite expensive) in particular the Breakwater Lodg= e Hotel. In fact we have made an agreement with the Breakwater Lodge Hotel; they are retaining some rooms for conference participants at a (mildly) reduced rate. If you wish to reserve a room there we can provide you with= a reference to include in your application for a reservation. The Waterfron= t is very attractive and a prime tourist destination with many good restaurants and shops and is situated in the working harbour about 15 minutes by car from UCT. If you prefer a hotel or B&B near UCT then you can make your own arrangements by consulting the web page http://www.mth.uct.ac.za/sams/accomodation.html which summarises options. (This page was constructed for another conference which is being hosted b= y UCT). If you hire a car then you need not restrict yourself to any of the options mentioned above. Note that if you are staying for an additional w= eek then you may choose to change accommodation in the second week to suit yo= ur purpose. REGISTRATION: Please email us the following information: 1. Full Name and Affiliation/Address (as you prefer it to appear in the l= ist of Participants): 2. Pleae indicate if you would you like to give a talk and if so the preferred duration. The final time allotted will be at the discretion of = the Scientific Committee. 3. Do you intend to stay on after the conference for an additional week? 4. Please indicate if you wish to stay at Breakwater Lodge. REGISTRATION FEE: The registration fee is ZAR 1500 South African rands which is approximate= ly 150 euro. Payment details will be in the third announcement. ABSTRACT SUBMISSION: Please send us your Abstract as a PDF file of not more than two pages bef= ore 1 December. PROCEEDINGS VOLUME: The final decision about the journal has not been made yet, but we expect= a good Proceedings Volume with papers being carefully refereed. PLEASE DO NOT HESITATE TO ASK ANY QUESTIONS Please direct all queries to George Janelidze at George.Janelidze@uct.ac.= za We look forward to seeing you in Cape Town. On behalf of the organising committee: John Frith, Christopher Gilmour, David Holgate, George Janelidze From rrosebru@mta.ca Tue Oct 23 09:06:46 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 23 Oct 2007 09:06:46 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IkIJU-00000T-Gl for categories-list@mta.ca; Tue, 23 Oct 2007 08:52:48 -0300 To: categories@mta.ca Subject: categories: Assoc. Prof. position at IT University of Copenhagen From: Lars Birkedal Date: Tue, 23 Oct 2007 11:32:12 +0200 MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 86 The IT University of Copenhagen invites applications for a tenured position as Associate Professor starting March 1, 2008 in the Programming, Logic and Semantics (PLS) group. The Programming, Logic and Semantics (PLS) group at the IT University of Copenhagen conducts research in the semantics of logics and programming languages; models for concurrent, mobile and distributed systems; logical frameworks, modular software verification; programming language implementation techniques; program analysis; and programming language technology for distributed and mobile applications, in particular for context-aware mobile computing. Application deadline is Nov. 12, at noon. Please see http://www1.itu.dk/sw70441.asp for the full official announcement. Best wishes, Lars Birkedal From rrosebru@mta.ca Wed Oct 24 15:34:01 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 24 Oct 2007 15:34:01 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IkksU-0000kp-TX for categories-list@mta.ca; Wed, 24 Oct 2007 15:22:51 -0300 Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed From: MFPS Subject: categories: MFPS 24 First Call for Papers Date: Wed, 24 Oct 2007 12:03:20 -0500 To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 88 Dear Colleagues, Below is the First Call for Papers for MFPS 24, which will be held on the campus of the University of Pennsylvania from May 22 through May 24, 2008, with a Tutorial Day on May 21. Best regards, Mike Mislove CALL FOR PAPERS MFPS XXIV Twenty-fourth Conference on the Mathematical Foundations of Programming Semantics University of Pennsylvania Philadelphia, PA USA May 22 - 25, 2008 Partially Supported by US Office of Naval Research The MFPS conferences are devoted to those areas of mathematics, logic, and computer science which are related to models of computation, in general, and to the semantics of programming languages, in particular. The series has particularly stressed providing a forum where researchers in mathematics and computer science can meet and exchange ideas about problems of common interest. As the series also strives to maintain breadth in its scope, the conference strongly encourages participation by researchers in neighboring areas. TOPICS include, but are not limited to, the following: biocomputation; categorical models; concurrent and distributed computation; constructive mathematics; domain theory; formal languages; formal methods; game semantics; lambda calculus; logic; non-classical computation; probabilistic systems; process calculi; program analysis; programming-language theory; quantum computation; rewriting theory; security; specifications; topological models; type systems; type theory. The Twenty-fourth Conference on the Mathematical Foundations of Programming Semantics (MFPS XXIV) will take place on the campus of University of Pennsylvania, Philadelphia, PA USA from Thursday, May 22 through Sunday, May 25, 2008. The Organising Committee for MFPS consists of Stephen Brookes (CMU), Achim Jung (Birmingham), Catherine Meadows (NRL), Michael Mislove (Tulane), and Prakash Panangaden (McGill). The local arrangements for MFPS XXIV are being overseen by Andre Scedrov (Penn). The INVITED SPEAKERS for MFPS XXIV are Samson Abramsky, Oxford Luca Cardelli, Microsoft Research, Cambridge Dusko Pavlovic, Kestrel Institute Benjamin Pierce, Penn Phil Scott, Ottawa James Worrell, Oxford In addition, there will be four special sessions: - A session honoring Phil Scott on the occasion of his 60th birthday year, which is being organized by Rick Blute (Ottawa) and Andre Scedrov (Penn). - A session on Systems Biology will be held in conjunction with Luca Cardelli's plenaary talk. It is being organized by Jean Krivine (LIX). - A third session will be devoted to Type Theory. It is being organized by Benjamin Pierce and by Robert Harper (CMU) will be held in conjunction with Benjamin Pierce's plenary talk. - The fourth special session will be on Security, and will be organized by Catherine Meadows (NRL) in conjunction with Dusko Pavlovic's plenary talk. Further, there will be a TUTORIAL DAY on May 21. The topic will be Category Theory and Its Applications to Theoretical Computer Science. It is being organized by Phil Scott (Ottawa); the speakers will be announced at a later date. This event will be free to all those who are interested in attending. The remainder of the program will consist of papers selected by the following PROGRAM COMMITTEE Andrej Bauer (Ljubljana), CHAIR Ulrich Berger (Swansea) Lars Birkedal (Copenhagen) Jens Blanck (Swansea) Steve Brookes (CMU) Bob Coecke (Oxford) Karl Crary (CMU) Martin Escardo (Birmingham) Achim Jung (Birmingham) Jean Krivine (LIX) James Laird (Sussex) Paul Levy (Birmingham) Catherine Meadows (NRL) Michael Mislove (Tulane) Catuscia Palamidessi (INRIA) Prakash Panangaden (McGill) Alex Simpson (Edinburgh) Christopher Stone (Harvey Mudd) Thomas Streicher (Darmstadt) James Worrell (Oxford) from submissions received in response to this Call for Papers. The CONFERENCE PROCEEDINGS will be published by ENTCS (Electronic Notes in Theoretical Computer Science ). Submission instructions, style files for preparing a submission, and a link to the MFPS XXIV submission site will be available soon on the conference web page: IMPORTANT DATES: * Fri Mar 7: Paper registration deadline, with short abstracts. * Fri Mar 14: Paper submission deadline. * Fri Apr 7: Author notification. * Fri Apr 21: Final versions for the proceedings. From rrosebru@mta.ca Wed Oct 24 21:31:43 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 24 Oct 2007 21:31:43 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IkqXv-0004Ur-H9 for categories-list@mta.ca; Wed, 24 Oct 2007 21:25:59 -0300 Message-ID: <00dd01c8167c$61a22070$0b00000a@C3> From: "George Janelidze" To: Subject: categories: Max Kelly Conference: in addition to SECOND ANNOUNCEMENT Date: Wed, 24 Oct 2007 22:28:04 +0200 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 89 In the Second Announcement for the Max Kelly Conference, among several suggestions for the accommodation, the following two places were specifically mentioned: 1. "All Africa House" (contact Callie at prezandt@protem.uct.ac.za to make a reservation and mention that you are a participant in the Max Kelly Conference). 2. "Breakwater Lodge Hotel" - which now gave us the following instruction: > The Guest must quote the reference # 384203 "The Max Kelly Conference" George Janelidze From rrosebru@mta.ca Thu Oct 25 18:28:46 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 25 Oct 2007 18:28:46 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1IlA2J-00045A-3D for categories-list@mta.ca; Thu, 25 Oct 2007 18:14:39 -0300 Date: Thu, 25 Oct 2007 12:11:33 -0400 (EDT) From: Michael Barr To: Categories list , Subject: categories: email address MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 90 For a while I was using email addresses either barr@barrs.org or mbarr@barrs.org. It turns out that the math dept has installed a really good spam filter that stuff sent to barrs.org avoids. So I ask you all to delete that address from your address books and use only barr@math.mcgill.ca Thanks in advance for this. M From rrosebru@mta.ca Sun Oct 28 20:16:29 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 28 Oct 2007 20:16:29 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ImH6c-0006sp-Pp for categories-list@mta.ca; Sun, 28 Oct 2007 19:59:42 -0300 Date: Sun, 28 Oct 2007 08:06:06 -0500 (EST) From: Michael Barr To: Categories list Subject: categories: Warning about Adobe 8 MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 91 Whatever you do, do not upgrade to Adobe reader 8. I found this on the texhax list. >>Has anyone else been clobbered by the discovery that Adobe Acrobat 8 >>tacitly suppresses all ligature glyphs of the fi, fl, ff, ffi, and ffl >>sort and displays blanks in their place. They do this without warning, >>so that a file which displays perfectly well in Acrobat 7 is made >>unreadable in Acrobat 8. >> It turns out that files converted (from the ps file) by the distiller (which costs something like $500) do not have this problem. I guess Adobe is tired of free use of their format. At TAC, we still consider the dvi to be the official format. Michael From rrosebru@mta.ca Sun Oct 28 20:16:30 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 28 Oct 2007 20:16:30 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ImH67-0006qV-33 for categories-list@mta.ca; Sun, 28 Oct 2007 19:59:11 -0300 Date: Sat, 27 Oct 2007 16:18:41 +0400 From: "Yuri Pritykin" To: "Yuri Pritykin" Subject: categories: CSR 2008: First Call for Papers MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 7bit Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 92 CSR 2008: First Call for Papers 3rd International Computer Science Symposium in Russia June 7-12, 2008, Moscow, Russia Organizers: Dorodnicyn Computing Centre of Russian Academy of Sciences, Institute for System Programming of Russian Academy of Sciences, Moscow State University, Moscow Institute of Open Education, Institute of New Technologies. Opening lecture: Avi Wigderson (IAS, Princeton). CSR 2008 is the third conference in a series of regular events started with CSR 2006 in St.Petersburg (see LNCS 3967) and CSR 2007 in Ekaterinburg (see LNCS 4649). It intends to reflect the broad scope of international cooperation in computer science. CSR 2008 consists of two tracks: Theory Track and Applications and Technology Track. Program committee of Theory Track: Sergei Artemov, Matthias Baaz, Boaz Barak, Lev Beklemishev, Harry Buhrman, Andrei Bulatov, Evgeny Dantsin, Volker Diekert, Anna Frid, Andreas Goerdt, Andrew Goldberg, Dima Grigoriev, Yuri Gurevich, Edward Hirsch, Nicole Immorlica, Pascal Koiran, Michal Koucky, Yury Makarychev, Yuri Matiyasevich, Alexander Razborov (chair), Victor Selivanov, Alexander Shen, Helmut Veith, Nikolai Vereshchagin, Sergey Yekhanin. Program committee of Applications and Technology Track includes Robert Bauer, Egon Boerger, Stephane Bressan, Gabriel Ciobanu, Maxim Grinev, Michael Kishinevsky, Gregory Kucherov, Alexandre Petrenko, Andreas Reuter, Anatol Slissenko (chair), Elena Troubitsyna, Andrei Voronkov, Sergey Zhukov. Conference chair: Alexei Semenov. Theory Track topics include * algorithms and data structures; * complexity and cryptography; * formal languages and automata; * computational models and concepts; * proof theory and applications of logic to computer science. Application Track topics include * artificial intelligence; * bio-informatics; * computer architecture, hardware design, nanotechnology; * databases and knowledge bases, information retrieval and search, Web technologies; * numerical and symbolic computing; * programming for parallel computing; * software development and software validation methods and tools. Submissions: Authors are invited to submit an extended abstract or a full paper of at most 10 pages preferably in the LNCS format. Proofs and other material omitted due to space constraints are to be put into a clearly marked appendix to be read at discretion of the referees. Papers must present original (and not previously published) research. Simultaneous submissions to journals or to other conferences with published proceedings are not allowed. The proceedings of the symposium will be published in Springer's LNCS series. Important dates: * Paper submission (via EasyChair): December 9, 2007. * Notification: February 8, 2008. * Symposium: June 7-12, 2008. Further information and contacts: Web: http://csr2008.ru/ Email: info@csr2008.ru -- CSR 2008 organizers, info@csr2008.ru From rrosebru@mta.ca Mon Oct 29 19:46:58 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 29 Oct 2007 19:46:58 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ImdGX-0004bp-5f for categories-list@mta.ca; Mon, 29 Oct 2007 19:39:25 -0300 Date: Sun, 28 Oct 2007 17:26:41 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: Categories list Subject: categories: Re: Warning about Adobe 8 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 93 Have you seen this problem yourself? I've been unable to duplicate it on either Linux or Windows XP, using free Acrobat Reader 8.1.1 dated August 20 on both platforms. I put all five ligatures in a latex file and compiled it directly to pdf with pdflatex, then as a double check indirectly with latex to dvi then to ps with dvips then to pdf with ps2pdf. On XP I did this with the latex that comes with Cygwin, on Linux with the latex that comes with Redhat FC4 (old system I haven't upgraded for a while). Is the problem independent of font? Of dvi-to-pdf converter? Of operating system? Etc, etc. Vaughan Michael Barr wrote: > Whatever you do, do not upgrade to Adobe reader 8. I found this on the > texhax list. > >>> Has anyone else been clobbered by the discovery that Adobe Acrobat 8 >>> tacitly suppresses all ligature glyphs of the fi, fl, ff, ffi, and ffl >>> sort and displays blanks in their place. They do this without warning, >>> so that a file which displays perfectly well in Acrobat 7 is made >>> unreadable in Acrobat 8. >>> > > It turns out that files converted (from the ps file) by the distiller > (which costs something like $500) do not have this problem. I guess Adobe > is tired of free use of their format. At TAC, we still consider the dvi > to be the official format. > > Michael > > > From rrosebru@mta.ca Mon Oct 29 19:46:58 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 29 Oct 2007 19:46:58 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ImdFu-0004Xe-6A for categories-list@mta.ca; Mon, 29 Oct 2007 19:38:46 -0300 From: Robert L Knighten MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Date: Sun, 28 Oct 2007 17:13:39 -0700 To: Categories list Subject: categories: re: Warning about Adobe 8 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 94 Michael Barr writes: > Whatever you do, do not upgrade to Adobe reader 8. I found this on the > texhax list. > > >>Has anyone else been clobbered by the discovery that Adobe Acrobat 8 > >>tacitly suppresses all ligature glyphs of the fi, fl, ff, ffi, and ffl > >>sort and displays blanks in their place. They do this without warning, > >>so that a file which displays perfectly well in Acrobat 7 is made > >>unreadable in Acrobat 8. > >> > > It turns out that files converted (from the ps file) by the distiller > (which costs something like $500) do not have this problem. I guess Adobe > is tired of free use of their format. At TAC, we still consider the dvi > to be the official format. > > Michael > What is the context? I've been using Adobe Reader 8 on Windows since it first came out and have never seen this, very definitely including pdf files created by pdflatex on both Windows and Linux. As a quick check I just created a number of pdf files in various ways on both Windows and Linux and viewed them in both places using both Adobe Reader 7 and Adobe Reader 8 and was unable to see any difference at all. -- Bob -- Robert L. Knighten RLK@knighten.org From rrosebru@mta.ca Mon Oct 29 19:46:58 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 29 Oct 2007 19:46:58 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ImdJ1-0004v6-5M for categories-list@mta.ca; Mon, 29 Oct 2007 19:41:59 -0300 Date: Mon, 29 Oct 2007 07:35:04 -0500 (EST) From: Michael Barr To: Categories list Subject: categories: Re: Warning about Adobe 8 MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 95 I have to admit that I never tested it, just copied the complaint from texhax, usually reliable. I just tried it on a file created by dvipdfm (I rarely use pdftex or pdflatex) and it seems fine. I have version 8.1.1. So perhaps it was an early bug, now corrected. But my main point--that we cannot tie our future to commercial software that the proprietor can change at will--remains valid. Michael On Sun, 28 Oct 2007, Robert L Knighten wrote: > Michael Barr writes: > > Whatever you do, do not upgrade to Adobe reader 8. I found this on the > > texhax list. > > > > >>Has anyone else been clobbered by the discovery that Adobe Acrobat 8 > > >>tacitly suppresses all ligature glyphs of the fi, fl, ff, ffi, and ffl > > >>sort and displays blanks in their place. They do this without warning, > > >>so that a file which displays perfectly well in Acrobat 7 is made > > >>unreadable in Acrobat 8. > > >> > > > > It turns out that files converted (from the ps file) by the distiller > > (which costs something like $500) do not have this problem. I guess Adobe > > is tired of free use of their format. At TAC, we still consider the dvi > > to be the official format. > > > > Michael > > > > What is the context? I've been using Adobe Reader 8 on Windows since it first > came out and have never seen this, very definitely including pdf files created > by pdflatex on both Windows and Linux. As a quick check I just created a > number of pdf files in various ways on both Windows and Linux and viewed them > in both places using both Adobe Reader 7 and Adobe Reader 8 and was unable to > see any difference at all. > > -- Bob > > From rrosebru@mta.ca Mon Oct 29 19:46:58 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 29 Oct 2007 19:46:58 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ImdIC-0004nT-8G for categories-list@mta.ca; Mon, 29 Oct 2007 19:41:08 -0300 Mime-Version: 1.0 (Apple Message framework v752.3) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: Categories list Content-Transfer-Encoding: 7bit From: Michael Mislove Subject: categories: Re: Warning about Adobe 8 Date: Mon, 29 Oct 2007 08:05:52 +0100 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 96 Mike, I think there may be some confusion here. Adobe Acrobat 8 costs about $450, and it will produce pdf files from various inputs. I assume at this price, it will produce files that Adobe Reader 8 displays properly. Adobe Reader 8 is free, but it only has the capability of displaying pdf files, not creating them. I wonder if it is Reader that's the culprit here, by not displaying the ligatures you list properly if the files are created by some method that doesn't use Acrobat. If so, there's no need to abandon pdf as a preferred format - just caution users to view them in Acrobat 7 until Adobe "fixes the problem" (sic). In particular, using the now- standard methods built into most (La)TeX distributions should still generate pdf files that will display correctly on most pdf apps - Preview.app for Macs and xpdf for Linux or other UNIX-based systems. No longer being a Windows user, I am not sure what alternative pdf file viewers are available for it, but this seems an ideal opportunity for someone to create one to fill an obvious need. In any case, using dvi as the preferred format has its drawbacks. Notably, it is binary, and hence can't be included in emails without extra effort. It also generates files that usually are considerably larger than corresponding pdf files, which makes sending them as email attachments a problem: most email servers now limit the size of attachments (the server at Tulane, which is admittedly more restrictive than most, simply throws such emails away, warning neither the sender nor the receiver), which forces one to place the files online for others to download them. In any case, I think more research is needed before a move like the one you are proposing for TAC is warranted. And, I'd be interested to know the exact nature of the problem. Best regards, Mike On Oct 28, 2007, at 2:06 PM, Michael Barr wrote: > Whatever you do, do not upgrade to Adobe reader 8. I found this on > the > texhax list. > >>> Has anyone else been clobbered by the discovery that Adobe Acrobat 8 >>> tacitly suppresses all ligature glyphs of the fi, fl, ff, ffi, >>> and ffl >>> sort and displays blanks in their place. They do this without >>> warning, >>> so that a file which displays perfectly well in Acrobat 7 is made >>> unreadable in Acrobat 8. >>> > > It turns out that files converted (from the ps file) by the distiller > (which costs something like $500) do not have this problem. I > guess Adobe > is tired of free use of their format. At TAC, we still consider > the dvi > to be the official format. > > Michael > > =============================================== Professor Michael Mislove Phone: +1 504 862-3441 Department of Mathematics FAX: +1 504 865-5063 Tulane University URL: http://www.math.tulane.edu/~mwm New Orleans, LA 70118 USA =============================================== From rrosebru@mta.ca Mon Oct 29 19:46:59 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 29 Oct 2007 19:46:59 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ImdK3-000535-AY for categories-list@mta.ca; Mon, 29 Oct 2007 19:43:03 -0300 Date: Mon, 29 Oct 2007 08:20:36 -0400 From: "Robert J. MacG. Dawson" MIME-Version: 1.0 To: Categories list Subject: categories: Re: Warning about Adobe 8 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 97 Michael Barr wrote: > Whatever you do, do not upgrade to Adobe reader 8. I found this on the > texhax list. > > >>>Has anyone else been clobbered by the discovery that Adobe Acrobat 8 >>>tacitly suppresses all ligature glyphs of the fi, fl, ff, ffi, and ffl >>>sort and displays blanks in their place. They do this without warning, >>>so that a file which displays perfectly well in Acrobat 7 is made >>>unreadable in Acrobat 8. >>> > > > It turns out that files converted (from the ps file) by the distiller > (which costs something like $500) do not have this problem. I guess Adobe > is tired of free use of their format. Well, I suppose that whether or not this is an accidental bug (and remember, as a famous corollary of Occam's Razor tells us, we should never put down to malice what can be adequately explained by stupidity) it will be a short enough time before somebody finds out how Distiller codes these glyphs and publicises it; and one upgrade after that before everybody's DVI->PDF utility follows suit. This does not strike me as a game that Adobe could play for long witout wrecking compatibility with their *own* software. Alternatively, one could presumably remap the glyphs so that Acrobat 8 didn't realize what it was displaying. > At TAC, we still consider the dvi to be the official format. Fair enough, though dvi has its own "intellectual property" problems with glyphs that the end user doesn't have a copy of. Not such a problem with TAC, I admit, but... -Robert From rrosebru@mta.ca Tue Oct 30 10:25:34 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 30 Oct 2007 10:25:34 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ImqvF-0006bl-Mj for categories-list@mta.ca; Tue, 30 Oct 2007 10:14:22 -0300 From: Gaucher Philippe To: Categories list Subject: categories: Re: Warning about Adobe 8 Date: Tue, 30 Oct 2007 08:58:19 +0100 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 98 Le lundi 29 octobre 2007 08:05, vous avez =E9crit=A0: > Mike, > I think there may be some confusion here. Adobe Acrobat 8 costs > about $450, and it will produce pdf files from various inputs. I > assume at this price, it will produce files that Adobe Reader 8 > displays properly. Adobe Reader 8 is free, but it only has the > capability of displaying pdf files, not creating them. I wonder if it > is Reader that's the culprit here, by not displaying the ligatures > you list properly if the files are created by some method that > doesn't use Acrobat. If so, there's no need to abandon pdf as a > preferred format - just caution users to view them in Acrobat 7 until > Adobe "fixes the problem" (sic). In particular, using the now- > standard methods built into most (La)TeX distributions should still > generate pdf files that will display correctly on most pdf apps - > Preview.app for Macs and xpdf for Linux or other UNIX-based systems. > No longer being a Windows user, I am not sure what alternative pdf As I already explained to Michael Barr, don't use Acrobat Reader. There exi= sts=20 other excellent pdf file readers. Under linux, xpdf, evince and kpdf (<--=20 excellent). Probably a lot of free pdf readers exist under other systems. S= ee=20 . You don't need to use= =20 acrobat softwares to produce and read pdf files. pg. From rrosebru@mta.ca Tue Oct 30 10:25:34 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 30 Oct 2007 10:25:34 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Imqtv-0006GT-Mj for categories-list@mta.ca; Tue, 30 Oct 2007 10:13:04 -0300 Subject: categories: Re: Warning about Adobe 8 Date: Tue, 30 Oct 2007 00:04:45 -0300 (ADT) To: categories@mta.ca (Categories list) MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit From: selinger@mathstat.dal.ca (Peter Selinger) Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 99 Hi Mike, in fairness, while Acrobat is commercial software, PDF is not really a proprietary format. The format has been fully and publicly documented since its inception in 1993. The PDF reference manual, like that of PostScript before it, is available from Adobe's website. As far as such technical references are concerned, it is also extremely accessible and well-written. I read the PostScript specification cover to cover, and I found it better than the reference manuals of most other programming languages. Adobe has extended the PDF specification from time to time (they are now at version 1.7). However, they have made an effort to remain backward compatible, and the changes in each version have been clearly and transparently documented. That is more than can be said of most commercial formats. According to their website, they intend to make version 1.7 into an ISO standard. Adobe should also be commended for keeping the PDF specification separate from the Acrobat implementation thereof. The specification is actually written in such a way that it allows arbitrary people to write applications that output PDF code, without having to use Distiller as a conduit. Regarding the reported problems with ligatures - if someone on that texhax list could produce a minimal actual example of a PDF file that displays incorrectly, it should be a relatively simple matter to match that against the PDF specification to determine whether the bug is in Acrobat or in the software that produced the file. Reference: PDF Reference, Sixth Edition, version 1.7 http://www.adobe.com/devnet/pdf/pdf_reference.html (Note: don't download the link called "PDF Reference and Related Documentation", because it requires - somewhat circularly - to be viewed with Acrobat Reader 8). -- Peter Michael Barr wrote: > > I have to admit that I never tested it, just copied the complaint from > texhax, usually reliable. I just tried it on a file created by dvipdfm (I > rarely use pdftex or pdflatex) and it seems fine. I have version 8.1.1. > So perhaps it was an early bug, now corrected. > > But my main point--that we cannot tie our future to commercial software > that the proprietor can change at will--remains valid. > > Michael > > On Sun, 28 Oct 2007, Robert L Knighten wrote: > > > Michael Barr writes: > > > Whatever you do, do not upgrade to Adobe reader 8. I found this on the > > > texhax list. > > > > > > >>Has anyone else been clobbered by the discovery that Adobe Acrobat 8 > > > >>tacitly suppresses all ligature glyphs of the fi, fl, ff, ffi, and ffl > > > >>sort and displays blanks in their place. They do this without warning, > > > >>so that a file which displays perfectly well in Acrobat 7 is made > > > >>unreadable in Acrobat 8. > > > >> > > > > > > It turns out that files converted (from the ps file) by the distiller > > > (which costs something like $500) do not have this problem. I guess Adobe > > > is tired of free use of their format. At TAC, we still consider the dvi > > > to be the official format. > > > > > > Michael > > > > > > > What is the context? I've been using Adobe Reader 8 on Windows since it first > > came out and have never seen this, very definitely including pdf files created > > by pdflatex on both Windows and Linux. As a quick check I just created a > > number of pdf files in various ways on both Windows and Linux and viewed them > > in both places using both Adobe Reader 7 and Adobe Reader 8 and was unable to > > see any difference at all. > > > > -- Bob > > > > > > > From rrosebru@mta.ca Tue Oct 30 10:42:46 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 30 Oct 2007 10:42:46 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1ImrLM-0002B3-6a for categories-list@mta.ca; Tue, 30 Oct 2007 10:41:20 -0300 Date: Tue, 30 Oct 2007 10:10:34 +0100 From: Andrej Bauer MIME-Version: 1.0 To: Categories list Subject: categories: Re: Warning about Adobe 8 Content-Type: text/plain; charset=ISO-8859-2; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 100 Michael Barr wrote: > But my main point--that we cannot tie our future to commercial software > that the proprietor can change at will--remains valid. Please do not confuse software and standards. PDF is an open standard, see http://en.wikipedia.org/wiki/Portable_Document_Format, which among other things means that Adobe cannot "take it away" from the public. They can take away their PDF viewer, but this is not an issue as there are a number of other viewers available. I use kpdf (a KDE incarnation of xpdf I believe), for example, and I am perfectly happy with it. As a bonus, kpdf does not display annoying adds in the upper right corner, which the free Adobe reader does. Best regards, Andrej From rrosebru@mta.ca Tue Oct 30 14:45:50 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 30 Oct 2007 14:45:50 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1Imv3y-0007Ms-Ck for categories-list@mta.ca; Tue, 30 Oct 2007 14:39:38 -0300 To: Categories From: JeanBenabou Subject: categories: Historical terminology,.. and a few other things. Date: Tue, 30 Oct 2007 10:02:52 +0100 Content-Transfer-Encoding: quoted-printable Content-Type: text/plain;charset=ISO-8859-1;delsp=yes;format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 101 Historical terminology,.. and a few other things. 0. Prologue 0.1. I want to thank all the colleagues who answered my questions on =20= "historical terminology". I shall try to answer all of them, and =20 shall greatly appreciate any remarks or comments about the present mail 0.2. I type very slowly, all the more so because I decided to =20 respect the constraints of "Outypo" in this mail.(See the appendix =20 for the meaning of this word). These constraints explain the unusual =20 presentation of this text 0.3. I shall abbreviate by (c), (cc),(lcc) and (lcct) the =20 following properties of categories: "cartesian", "cartesian closed", =20 "locally cartesian closed" , and "locally cartesian closed with a =20 terminal object". 0.4. I shall denote by El the book of P. Johnstone: Sketches of an =20 Elephant, Vol.1 and, and for references I shall use the monumental =20 bibliography of El in the following manner: (i) If the reference is there, I use El then the number, in bold =20 face between brackets, e.g.(El 911) means : S.C. Nistor and I. Tofan, On the category ab(E) (Romanian), =20 Bul.Inst.Politechn. Iasi (I) (1985) 205-207 ; MR 87g:18001 (ii) If the reference is not there, I shall write the symbol (El ?) =20= followed by the minimum information permitting to localize it, e.g. (El ?) P.J. Cohen, Set theory an the Continuum Hypothesis 0.5. The notations shall be either standard or self explanatory with =20 the following exception: I shall abbreviate "x is an element of y" by =20= "x@y". For example in the following definition: A category X is lcc if it satisfies: forall x (x@Ob(X) =3D> X/x is cc) 0.6. A first draft had convinced me that my mail would be abnormally =20 long, so I have decided to break it in two parts. The first, almost =20 totally devoted to (lcc), already very long, which I send =20 immediately. I shall then wait a few days for the second part, =20 devoted to (c) and (cc). This will permit me to receive any remarks =20 or comments you would want to make about this first part, and to =20 answer them as well as I can in the second part. I hope I won't be too boring, and apologize in advance if I am 1. Locally cartesian closed categories (lcc) =46rom Prof. Peter Johnstone's answer to the question: Why did he =20 impose a terminal object in his definition of lcc, I quote: "I did that because it seemed the appropriate convention to adopt in the context of topos theory. I wasn't trying to dictate to the rest of the world what the convention should be. On the other hand, there seem to be remarkably few `naturally occurring' examples of locally cartesian closed categories which lack terminal objects: the category of spaces (or locales) and local homeomorphisms is almost the only one I can think of." Since "he can think" only of very few or almost only one "naturally =20 occurring" lcc's which lack terminal objects, let me provide "a lot" =20= more ones. 1.1 First examples of categories which are l.c.c but have no =20 terminal object (i) The smallest "I can think of" namely O, the empty category. (ii) A little bit bigger, but not too big: 1+1, the discrete category =20= with two objects. (iii) Any discrete category ( except 1 of course) (iv) Any group (again except 1). (v) Any groupo=EFd (non trivial, i.e. non equivalent to 1). The reason for (v) is the following characterization of groupoids, =20 which does not seem to have found its way in standard texts on =20 category theory: A category X is a groupoid iff all its slices are equivalent to 1 Moreover, if X is a groupoid, the following are equivalent : (a) X has a terminal object (b) X is non empty and has binary products (c) X is equivalent to 1 Since (i), to (v) are groupoids, I shall count the whole previous =20 list as one single type of example, and give others which don't fit =20 in this pattern. (vi) The ordered set Omega of natural numbers. (viI) More generally any limit ordinal . (viii) The ordered category On of all ordinals. (ix) Any coproduct of l.c.c. (Even if they all have terminal objects, =20= provided there is more than one such category) (x) Any rooted tree, with the canonical ordering obtained by taking =20 the root as smallest element. In particular the binary tree. And of =20 course, by (ix) any "planted forest", i.e. coproduct of rooted trees. (xi) the ordered set R_+ of non negative real numbers. (xii) A little bit bigger than Omega, but still countable namely: the =20= category with objects the finite cardinals and maps the injections. (xiii) The following one might be "handy" for logical purposes: If C is a topos, the category Mono(C) having the same objects, and =20 as maps the monos of C is lcc without terminal object. Question: Which, among the previous list, is so "unnaturally" =20 occurring" that it should be "banned" from category theory? And =20 moreover at the cost of a "linguistic violation" of the commonly =20 adopted meaning of "local"? 1.2. First "stability" results The following theorems "explain" and generalize many of the previous =20 examples, and permit to construct many more important lcc's without =20 terminal objects. 1.2.1 Theorem If C is a lcc category, so is Mono(C). 1.2.2 Theorem Let P: D-->C be a discrete fibration, if C is l.c.c. =20= so is D 1.2.3 Corollary Let C be a category and D be a sieve of C. If =20= C is lcc so is D. I shall leave to your "fertile brains" (Preface of El , p.viii) the =20 pleasure to use 1.2.1 and 1.2.2 to construct lcc's without terminal =20 objects, and I shall use only the much weaker 1.2.3 to give more =20 examples. They all fit in the following general pattern: Start with a =20= C which is lcc, and may or may not have a terminal object. Suppose is =20= is equipped with a notion of "boundedness",and take for D the sieve =20 of bounded objects. Here are a few important examples: (xiv) Take for C an elementary topos, and call an object of C =20 bounded if it is contained in a (Kuratowski)-finite object. (xv) Let A be any small category. Take for C the category A^ of =20 presheaves over A and call a presheaf bounded if it is contained in a =20= representable one. (xvi) Let X be a metric space. Take C=3DOpn(X) , the locale of its open =20= subsets, and call an open set U bounded if it is contained in a sphere. (xvii) Let X be a topological space, and again C=3DOpn(X), and call U =20= bounded if it is contained in a compact. (xviii) one can "sheafify" (xvi) and get the lcc of sheaves with =20 compact support. How many more examples does one need? Let me mention that the direct "proof" of 1.2.3 is so simple you =20 could cry: If D is a sieve of C, all its slices are slices of C, thus =20= any local property, not only lcc, satisfied by C will be shared by D. =20= But, even if C has a terminal object, D does not need to have one. 1.2.4 Remarks "en vrac" 1- Even if we assume in the previous results that C has a terminal =20 object, neither Mono(C) nor D need have one. Thus (lcc) has many more =20= "stability properties" than (lcct). See also 1.1 (ix) 2- It is even possible to have an lcc category , with binary =20 products but no terminal object. This is the case in the examples =20 (xv), (xvi) and (xvii) if the whole space X is not =20 "bounded" (partial answer to Dubuc) 3- There even exits an lcc which has binary products and is not =20 connected, namely 0 . (Dubuc again) 4 - There are much stronger " stability results" than the previous =20 ones permitting to construct lcc's but it would take too much space =20 to build the the "set up" where they can all be stated, let alone =20 proved. I shall say a few words about parts of this set up in the =20 next section, and show how it can be used for "general" local =20 properties, not just lcc. Question: What important theorems hold for (lcct), but not for (lcc)? If you ask the question in the other direction my answer is easy: All =20= the stability theorems I mentioned Remark: In the theory of fibered categories we work with fibers and =20 slices and the introduction of unneeded terminal objects weakens the =20 results or confuses some issues, and sometimes does even both. 1.2.5 Terminology again. There is by now an unwritten but (almost) =20 unanimous "linguistic consensus" on the following terminology: If P =20 is a "property" of categories, a category satisfies P locally iff all =20= its slices satisfy P. I do not want to "dictate" anything to anybody, but I think we could =20 all easily agree on this "unifying" terminology and, if for some =20 mathematically imperative reason, we need a different notion, we =20 should use our imagination to give it another name. There is already an unfortunate exception, namely: locally small =20 categories, which I never liked, (I would have preferred something =20 like: piecewise small). But "locally small" has been used by =20 countless mathematicians in countless texts, and in such a case I am =20 not very prone to change the terminology, even if I don't like it. 2 . Local properties of functors We extend the previous definitions to properties of functors in the =20 following manner. 2.1 Basic definitions 2.1.1. If F: X-->Y is a functor, and x is an object of X, the =20 slice of F at x is the obvious functor F/x: X/x-->Y/Fx induced by F We remark that, for G: Y-->Z , we have: (GF)/x =3D (G/F(x))(F/x) . 2.1.2. A property (of functors) is a class P of functors satisfying: Every iso F: X-->Y is in P, and P is stable under composition 2.1.3. If P is a property, a functor is locally in P if all its =20 slices are in P , thus we get a new property called the localized =20 of P and denoted by L(P) 2.1.4. If P is a property we say that a category X is in P if the =20 functor X-->1 is . In that case every category isomorphic to X is =20 also in P. 2.1.5. Properties are ordered by inclusion, we shall write P=3D>P' =20= for " P is contained in P' ". This implies L(P)=3D>L(P') 2.2 Remarks, again "en vrac" 1- One can define more general notions of "localness" than those I =20 gave in 2.1, but I shall not mention them in this mail 2- =46rom a categorical point of view the definitions I gave are =20 "natural" . They are also very simple and, because of this =20 simplicity, one might think that not much can be derived from them. =20 That it need not be so, comes from the fact that very simple =20 properties may have much "richer" localizations. 3- Although this may seem "strange", I do not assume that: if F: X-->Y is in P, and F': X-->Y is isomorphic to F, then F' =20 is in P, let alone that every equivalence functor is in P. e.g. P could very =20 well be the property of functors which are surjective on objects. .2.3 Local Iso's I denote by Iso the smallest property: the only functors in it are =20 iso's, the functors in its localized are thus local iso's. Quite =20 "surprisingly" we have: 2.3.1 Theorem A functor F: D-->C is a local iso iff it is a =20 discrete fibration As a consequence we have following "ubiquity" of discrete fibrations, =20= if F: D-->C is such a fibration, then for any property P, F is in =20= L(P). In particular, If C is in L(P) is in L(P) so is D. Since for any category X the functor 0-->X is a discrete =20 fibration, it will be be in L(P), and in particular so will be the =20 initial category 0. More than 2/3 of the examples of =A71 follow from the previous remarks =20= applied to the very special case of lcc's 2.4 Local surjective equivalences The smallest property "I can think of", after Iso of course, is SEq , =20= surjective equivalences, which I define as:functors which are =20 full,faithful and surjective on objects. (They are also surjective on =20= maps). Thus its localization L(SEq) is defined. Even more =20 "surprisingly", we have: 2.4.1 Theorem Let F: D-->C be a functor. The following are equivalent : (i) F is locally a surjective equivalence, i.e. F is in L(SEq) . (ii) Every map of D is F-Cartesian, and F is a fibration. (iii) F reflects isos and F is a fibration. (iv) F is a groupoid - fibration (i.e. a fibration where all the =20 fibers are groupoids. Almost every property P contains all equivalences, and in particular =20 the surjective ones. Thus we get an "almost ubiquity" of groupoid fibrations :For any =20 such P we have: (i) Every groupoid fibration F: D-->C is in L(P). In particular: (ii) If C is in L(P) so is D (iii) If G is a is a groupoid it is in L(P) This explains, and considerably generalizes 1.1 (iv), and it might =20 interest Ronnie Brown. 2.4.2 Remarks Surjective equivalences are much better than mere equivalences because : (i) They are stable under pull backs whereas equivalences are not, =20 and they can even be characterized as the only equivalences stable =20 under pull-backs. (ii) They are fibrations and, as fibrations , they have simple =20 characterizations, namely: Let F be a fibration: F is an equivalence <=3D> each fiber is equivalent to 1 <=3D> F is in = SEq (iii) They can be internalized in any regular category, and "behave" =20 there as well as one might want. (iv) I don't know who observed first that equivalences are not stable =20= under pull-backs. It is mentioned explicitly in Grothendieck's first =20 paper on fibrations, together with the fact that, for fibrations, =20 they are stable: (El ?) A. Grothendieck , Categories fibrees et descente; SGA 1961 . (v) Much later, Freyd and Scedrov (loc. cit. 1.361 p.19) have =20 considered surjective equivalences under the name of inflations, but =20= they were mainly interested, as most people, in "mere" equivalences =20 and the inflations served only to decompose equivalence functors. =20 In particular they never mentioned that inflations were fibrations, =20 nor that they were stable under pull-backs, both facts which are =20 important to me. 2.5. The local game Given time and space,I could have added to 2.3. and 2.4 a very long =20 list of localized properties, I shall just give an idea of the "local =20= game" I have been playing, with interruptions,for more than 25 years. =20= Chose simple properties P and find out what L(P) is. In the other =20 direction, chose some important (for you at least) property Q and =20 try to see if it is of the form L(P) for some P . Such a P need not =20 be unique, so try to find one "as simple as possible". 2.5.1. Example If P is the property of functors which are surjective =20= on objects and have a right adjoint, we have: A functor is in L(P) =20 iff it is a fibration. Thus, if F:X-->Y is a fibration, for every property Q such that =20 P=3D>Q , we have: If G:Y-->Z is in L(Q) so is GF, and in particular, if Y is in L(Q) =20= so is X. 2.5.2. The "game" can be made more complex, and more difficult, by =20 imposing further "constraints" (see appendix) of the kind: For which P's is L(P) stable under pull-backs, or pseudo pull backs, =20 or has "adapted" calculus of fractions, etc. Before I got used to the game, I had many "surprises". Each of them =20 brought a new result or at least a better understanding of old ones. 3. A few comments on the answers I received 3.1 General picture There seems to be a general agreement in all the mails about the fact =20= that (lcc) should not include terminal objects, except of course for =20 Prof. Peter Johnstone, but I have already made long comments about =20 his mail, and if he is not convinced by my purely mathematical =20 arguments, I'd greatly appreciate if he could tell me why he isn't. =20 Of course again: only for mathematical reasons. The mails which agreed with my definition explained this agreement =20 by two kind of arguments: (i) Purely linguistic and coherent use of "local". This includes Fred =20= Linton, Eduardo Dubuc and Phil Scott's second mail. As I totally =20 agree with them on this basis I shall make no more comments (ii) Arguments coming from "logical systems" such as dependent =20 types, lambda calculus, etc. This includes Phil Scott, Vaughan =20 Pratt,and Paul Taylor. I must confess i am not convinced by these =20 types of arguments, and I shall explain why. I expect strong =20 reactions to some of my statements, and I shall be very happy to to =20 hear them, and try to answer them. 3.2. First comments I'm absolutely sure, even if I don't know some of them very well, =20 that the "logical systems" mentioned in the previous sub-section, =20 provide very important examples of lcc or lcct categories. =20 Nevertheless I'm tempted to say So What? I won't say it because some people might think there are limits to =20 heresy, but very deep in my mind I'll continue to think it. Why? 3.2.1. Other important examples can arise from different domains of =20 Category Theory or more generally from mathematics. I gave a long list in =A71 , which I could easily have made much =20 longer, and even if someone could prove that, say, Omega , R_+ , any =20 groupoid, sheaves with compact support , could be described in terms =20 of "dependent types" or "lambda-calculus" I would still not be =20 convinced! Because each description would require a different ad-hoc "logical =20 system", which would certainly appear artificial to the specialists =20 of the domain, whereas for a categoricist these examples are all easy =20= and meaningful. In particular I ask the question: Which of the previous examples has ever appeared in the context of =20 such logical systems? 3.2.2. I shall go a bit further. Even If someone did come up with a =20 (meta) theorem of the kind: "Every lcc category can be described as a =20= suitable model of such a forma system" , (Which is probably true, =20 and probably easy to prove), even in that case, I would still not be =20 convinced! Because: How would one interpret the various stability theorems I =20 mentioned? For example: Suppose I know that a specific lcc category =20 C is a model of some formal system (F). By 1.2.1 and the, yet =20 unproved, meta-theorem I alluded to, I shall know that Mono(C) is a =20 model of another formal system of the same kind, say (F'). But then =20 how do I deduce syntactically (F') from (F) ? I contend that Category Theory by asking such natural questions, =20 might inspire some interesting formal constructions in various =20 "logical systems" 3.2.3. Category Theory is "irrigated" by many mathematical fields. An =20= important one is so called "categorical logic", but it is not the =20 only important one. And if we are tempted to think that some axioms =20 we assume on categories make them "the embodiments" of some kind of =20 logical system, they are almost never only that e.g. : Think of a topos as "semantics for intuitionistic formal =20 systems" (El. Preface) what is the "syntactic counterpart" of the =20 following well known and important result: If C is an internal category of a topos E, the category E^C is a =20 topos. Such a syntactic description could perhaps be given, but at what =20 cost? And would it clarify or obscure this basic result? 3.2.4. There is yet another question to specialists of "logical =20 systems" Any category with a terminal object is connected as a =20 category. lcct categories are connected. Now an elementary topos or a =20= locale are lcct categories , where there is another important notion =20 of connectedness, namely 1 is a connected object (I apologize for =20 such trivialities). Here is the question: Is there a similar notion of connectedness for the kind of "logical =20 systems" I mentioned earlier? 3.2.5. Suppose E is an elementary topos.(the assumption is much too =20= strong, but I make it to be on the safe side) One might want to =20 define internal categories in E which are lcc. Incidentally I did =20 it . It was easy. And in order to do it, I didn't have to =20 "internalize" (whatever that might mean) "Dependent Type Theory", or =20= any other logical system. 3.3. The main objection 3.3.1 I have given in 3.2. many reasons why arguments coming =20 exclusively from "categorical logic" did not convince me, but there =20 is a fundamental one, namely: Viewing some categories as embodiments of "logical systems", most of =20= the time does not give any indication about what the morphisms =20 between such categories ought to be, and sometimes even suggest wrong =20= directions. When we need these morphims,and in general we do need =20 them, the ultimate choice comes from mathematics, not logics. 3.3.2. First examples (i) What is the notion of morphism, if any, suggested by formal =20 systems such as : "Dependent Type Theory", or "Typed Lambda calculus" ? (ii) Has anybody defined a notion of morphism between =20 "Hyperdoctrines", and, if nobody has, why not? (iii) In =A72.5. of their very nice book (El 381) Freyd and Scedrov =20 define the notion of congruence on an Allegory. This a natural =20 definition of "syntactic type". But very quickly they restrict their attention to "amenable congruences" which are no =20 longer "syntactic". This is a typical illustration of the fact that =20 "ultimately, the choice comes from mathematics". (I shall come back =20 to this notion of congruence in the second mail) 3.3.3. A morphism of toposes is ...? In the preface of El one can find a very illuminating list of =20 "descriptions" beginning by "A topos is.." and numbered from (i) to =20 (xiii). I shall use the same numbering if I want to refer to some of =20 them. I was very much impressed, especially since it took more than =20 20 years to complete that list, and the contribution of, I quote: =20 "the category-theory community", and, "the theoretical computer =20 scientists" . Since there was not a single description of geometric morphisms, I =20 studied carefully that list, in the light of 3.3.1, to see which of =20 the 13 descriptions were most suitable to give indications about how =20 to describe these morphisms. Obviously (v) and (viii) are too =20 "sophisticated" and require too much preliminary knowledge of Topos =20 Theory, and many other domains, to serve my purpose, so I dropped =20 them and concentrated on the 11 remaining descriptions. And there, I had a big "surprise" : I am no linguist, and moreover =20 English is not my mother language, but I remarked that the "A" in "A =20 topos is" had different meanings, e.g. 1. In (xii) "A topos is" means "some toposes are". 2. In (i), (vi) and (xi) "A topos is" means "every Grothendieck =20 topos is" 3. In (ii),(iii), (iv),(vii),(ix),(x) and (xiii) "A topos is" means =20 "every topos is" I shall not insist on this "logical ambiguity",but obviously, if we =20 seek a general description of morphisms of toposes, we won't find it =20 in 1 because only "some" toposes fit in this description. In the "sublist" 3, (iii), (iv), (vii), and (xiii) come from various =20= logical systems. I have tried to figure out, thinking only in terms =20 of such systems, what a morphism should be. I confess I couldn't find =20= a natural definition of such morphisms between, say: two "..=20 (embodiments of) an intuitionistic higher-oder theory" , (iii), let =20 alone between one such embodiment and "..a setting for synthetic =20 domain theory" (xiii). I'm sure a helpful colleague will supply a =20 "bridge" between the two. The best I could do was to "describe", very =20= vaguely, logical morphisms between two toposes, and only when their =20 two "descriptions" were given by the same number on the list. When the "fertile brain" of Grothendieck (El preface, p.viii) gave =20 the definition of geometric morphism, he knew only (i) in "the =20 list", because he happened to have invented it. The definition was =20 given for purely mathematical reasons. And as all very deep =20 mathematical definitions, it has resisted time. It has even =20 anticipated time, because it is suitable for elementary toposes which =20= didn't even exist when he gave his definition! There is much more one can say about based toposes than about "mere" =20 toposes, and I'd be curious to know how specialists of "logical =20 systems" or "computer scientists" would have, even in a "descriptive" =20= manner, answered the question: A based topos is ... ? I have some comments, questions, and even a few answers, about "La =20 Lista", which is supposedly the fruit of: the category-theory =20 community and the theoretical computer scientists. But I shall =20 postpone them until "better times" 3.3.4. Logical categories and categorical logic I think that the scope of "categorical logic" should be much wider =20 than the mere study of the categories which "embody logical systems", =20= which I propose to call by the "generic" name of logical categories. =20 (But of course I would never dream of trying "to dictate" anything =20 to anybody, let alone to "the rest of the world") It could include in particular: (i) the study of "local properties" , a "flavor" of which has ben =20 given in =A72, (ii) Calculus of fractions, adapted to various "properties" of =20 categories and functors , a first part of which, with clear =20 motivations, can be found my paper (El 103) . But that was in 1989, =20 almost prehistory. Some of you may have doubts about the relevance of =20= this calculus of fractions to "categorical logic". In the second =20 part, if I don't have to answer too many questions or objections =20 about the present mail, in the same spirit as in =A72 I shall give a =20 few mathematical results to try to convince them (iii) abstract notions of homotopies in categories but also of =20 categories, i.e. in Cat, as defined by Grothendieck in his "Pursuing =20 stacks". A group of mathematicians, mostly French, are developing =20 his ideas, and for those who might be interested, apart from their =20 numerous papers, I recommend volumes 301 and 308 of "Asterique" : G. Maltsinotis, La theorie de l'homotopie de Grothendieck D.C. Cisinski, Les prefaisceaux comme modeles de l'homotopie This list is very far from complete, and I'm sure that many of you =20 have in mind some important parts of category theory which could be =20 added to it 3.3.5. I have worked, on and off, for more than 20 years, on some =20 aspects of this "categorical logic". I have talked two or three times =20= about my ideas and my very very first results, but I met only "polite =20= but indifferent" reactions. Maybe my work didn't deserve much more. =20 I don't care. Because, if you allow me to be a bit "personal", this =20 work has given me a lot of pleasure. In particular because it has =20 permitted to deepen my relation with old and dear friends such as =20 fibered categories, cartesian maps and functors, categories of =20 fractions, etc, and to improve my understanding, and knowledge, of =20 their "qualities". And, last but not least, to prove new mathematical =20= results about them. 3.4. The special case of Mr. Paul Taylor The answer to Paul Taylor deserves a special treatment. Although I =20 deeply regret it, it will not be only mathematical, but such a choice =20= was his to begin with. I quote his mail: "I am sorry to say that I have seen papers emanating from respectable universities in which the authors have appeared to believe that this is the definition. (One of the papers that I have in mind cites many eminent categorists, who may perhaps have an opinion about =20 having their names appear alongside a lot of complete nonsense.)" 3.3.1- Why? Why such petty and spiteful attacks on unnamed mathematicians, =20 without any proof or justification, in a purely "historical", non =20 polemic discussion? (c.f. The answers of all the other participants) Why didn't anybody react to such attacks, or to previous ones, by the =20= same "Mr" Taylor? Does he enjoy some "special status", or shall we =20 have to consider in the future such behaviors as "normal"? I quote him again: "My footnote refers to "other authors" who said that LCCCs should =20 have binary products; I think I may have had Thomas Streicher in =20 mind, but I don't recall what he may have said or in what paper." 3.3.2- Streicher &... others? Why mention Thomas Streicher without at least trying to find out what =20= he said or wrote precisely on the question? Why not mention P. Johnstone's "Elephant" where this is precisely =20 written, long after Taylor's world famous "footnote" was published. =20 Lack of courage? Fear for future promotions? Why not mention two other " eminent categorists" who made the same =20 mistake in a published paper that he certainly knew, namely Phil =20 Scott and...Paul taylor himself who wrote in their joint paper (El.=20 977) at the very first page in the fist proposition: "Let C be a locally cartesian closed category( that is, C has finite =20 limits and for each object X in C, the slice category category C/X is =20= cartesian closed)..." Lack of memory? Lack, again, of elementary courage or decency to =20 "confess past errors"? When was Mr. Taylor "struck by the light" ? When did he abandon the finite limits before writing "his footnote", =20 and as all new zealots, started condemning very strongly his former =20 "sinning colleagues"? Except the powerful ones, of course! 3.3.3- "Consensus"? I quote him again: "I confess that I'm a bit surprised to find that the consensus agrees with me, so to set matters straight I should also point out that my argument applies equally to elementary toposes and other familiar structures of categorical logic." I am greatly honored to find that I agree with Mr. Paul Taylor's =20 footnote in "his book", which I have not read, and have no intention =20= to read, about matters I had completely settled more than 20 years =20 before "the" book was published ! Mr. Taylor was answering me. Thus I very gratefully thank him for =20 teaching me a few things that presumably I didn't know such as: "The simplest formulation is that an LCCC is a category every slice of which is a CCC. In particular, every slice has binary products, which are pullbacks in the whole category." "Objects of an LCCC and the slices that they define correspond to objects of a base category and the fibres over them in a fibred or indexed formulation of logic," I certainly do agree, except on a "minor detail": I do not like the =20 idea that my name could be in any manner whatsoever associated with =20 "indexed categories". I never used the term, I said and wrote =20 countless times that I considered the notion as wrong. Maybe I am =20 wrong, the future will decide. But I want no part of responsibility =20 in the propagation of this notion. This is my choice as a =20 mathematician. Incidentally, I am in very good company, most =20 mathematicians, some of them outstanding use fibrations. Of course, I =20= am quite ready, if I am asked, to give,once again, purely =20 mathematical reasons for this choice. But I'm afraid it will, again, =20 be in vain, because : "Il n'est pire sourd que qui ne veut entendre". 4. Appendix: Outypo (for my friend Jacques Roubaud, a poet, a mathematician and an =20 innocent victim) 4.1 Many of you have probably heard of "Oulipo", Ouvroir de =20 Literature Potentielle, a literary group created in 1960 by Raymond =20 Queneau and Francois Le Lionnais. It proposed to create literary =20 texts submitted to well chosen but otherwise arbitrary =20 "constraints" , of various nature: linguistic, syntactic, =20 combinatorial, and even topological. (One of the best known examples =20 is due to Georges Perec who managed to write a whole,and good, novel =20 without ever using the letter "e" which is by far the most frequently =20= used in French). Jacques Roubaud, a member since 1966, has invented dozens of such =20 constraints, some of them quite sophisticated, using e.g non trivial =20 groups of permutations or topology "a la" Moebius strip or Klein =20 bottle. He is world wide known as the author of more than a dozen =20 novels, many thousands poems, and one of the best specialists of the =20 sonnet. (For more details, you can consult Wikipedia, about two "items": =20 Oulipo, and Jacques Roubaud) 4.2. In all my life I have written only three joint papers. The =20 first was: ( El ? ) J.Benabou, J.Roubaud, Monades et descente 4.3. Oulipo has "swarmed" from literature to many other domains : =20 painting, music, photography,etc. New groups have been created, all =20 over the world, in these domains. And, to "remember their =20 filiation", they have chosen their name according to the following =20 "constraint " : Ou X po , i.e; three syllables, the first "Ou", the =20 last "po", the "X" in the central one being an abbreviation of the =20 name of their domain. e.g. in the domain of painting, "peinture" in =20 french, there has existed for almost 20 years now "Oupeintpo" as: =20 Ouvroir de peinture potentielle. Before formally adopting "Outypo", I have consulted my "expert", =20 J.Roubaud who confirmed that the name was correctly formed, and that =20 my constraints were genuinely of "oulipian" nature. 4.4. Easy "oulipian" questions (i) Complete the reference of our joint paper (ii) Why, and of whom is Jacques Roubaud a victim? (iii) What does "Outypo" stand for, and what are its constraints ? From rrosebru@mta.ca Wed Oct 31 15:39:18 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 31 Oct 2007 15:39:18 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1InIEt-0002gE-82 for categories-list@mta.ca; Wed, 31 Oct 2007 15:24:27 -0300 Date: Wed, 31 Oct 2007 16:20:30 +0100 From: Uwe Egbert Wolter MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Comma categories Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 102 Dear all, I'm looking for a comprehensive exposition of definitions and results around comma/slice categories. Especially, it would be nice to have something also for non-specialists in category theory as young postgraduates. Is there any book or text you would recommend? Best regards Uwe Wolter From rrosebru@mta.ca Wed Oct 31 15:39:18 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 31 Oct 2007 15:39:18 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1InIEO-0002Xk-7S for categories-list@mta.ca; Wed, 31 Oct 2007 15:23:56 -0300 Date: Wed, 31 Oct 2007 07:53:13 -0500 (EST) From: Michael Barr To: Categories list Subject: categories: More on Adobe reader 8 MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 103 A man named Pierre McKay, one of the real tex gurus has now written that a large archive of files rendered into pdf by Ghostcript are now unreadable in Reader 8. My main point is that dvi should remain our ultimate format. Even when the viewers do not render the dvi file correctly, all the information is there and can be converted to pdf by whatever converter we have available. And authors should be encouraged to provide, if at all possible, files that do render properly in dvi format. This means avoid rotated letters and avoid, I much regret to say, diagram packages that work properly only in ps or pdf format. One of the things that makes xypic (not to mention packages based on it) so remarkable is that it manages to do all it does without using such tricks. Michael From rrosebru@mta.ca Thu Nov 1 10:42:16 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 01 Nov 2007 10:42:16 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1InaEl-0002cV-Vq for categories-list@mta.ca; Thu, 01 Nov 2007 10:37:32 -0300 Date: Wed, 31 Oct 2007 14:35:08 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: Categories list Subject: categories: Re: More on Adobe reader 8 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 104 Mike may have a point here. dvi has the advantage over pdf that Latin has over English, it's not evolving as fast. (Not that Latin didn't evolve: in two millennia it went from classical to medieval and ecclesiastical, the latter being today the official language of the Vatican City.) Of the sequence .tex -> .dvi -> .ps -> .pdf, it's a nice question which is the most appropriate archival medium. Perhaps the best strategy for future archivists is to archive all four, which already seems to be becoming the custom on some archives. However there should be a consensus as to what defines each of these formats. How do we know that Ghostscript-produced pdf is the same format as what pdflatex or ps2pdf or dvipdfm produces? It's relatively easy to get a pdf distiller to work with current releases of pdf readers, it is harder to get it to adhere to standards designed to survive upgrades of those readers, which may be the root cause of the present incident. How to rescue archives whose format has become obsolete, whether through using a nonrobust distiller or some other cause of software rot, is then an excellent question. The idea that we should all stick to Adobe Acrobat Reader 7.0 in perpetuity somehow doesn't sound optimal---what pdf's produced in 2017 will 7.0 be able to read? Vaughan Michael Barr wrote: > A man named Pierre McKay, one of the real tex gurus has now written that a > large archive of files rendered into pdf by Ghostcript are now unreadable > in Reader 8. My main point is that dvi should remain our ultimate format. > Even when the viewers do not render the dvi file correctly, all the > information is there and can be converted to pdf by whatever converter we > have available. And authors should be encouraged to provide, if at all > possible, files that do render properly in dvi format. This means avoid > rotated letters and avoid, I much regret to say, diagram packages that > work properly only in ps or pdf format. One of the things that makes > xypic (not to mention packages based on it) so remarkable is that it > manages to do all it does without using such tricks. > > Michael > > From rrosebru@mta.ca Thu Nov 1 10:42:16 2007 -0300 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 01 Nov 2007 10:42:16 -0300 Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1InaHS-00031w-JU for categories-list@mta.ca; Thu, 01 Nov 2007 10:40:18 -0300 Mime-Version: 1.0 (Apple Message framework v752.2) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Content-Transfer-Encoding: 7bit From: JeanBenabou Subject: categories: Re: Historical terminology,.. and a few other things. Date: Wed, 31 Oct 2007 23:53:07 +0100 To: Paul B Levy Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 106 Dear Paul, There are unfortunately TWO conflicting uses of "locally" in category theory, which have nothing to do with each other: One means "slicewise", which I was referring to, and the other means "homwise", coming from enriched categories. When we say lccc, we obviously refer to the first one. It was in order not to introduce further ambiguities in the FIRST notion that I wanted to get a "consensus" about IT. As for "indexed" versus "fibered" I have many times mentioned the PURELY MATHEMATICAL reasons of my preference. Here is a "test" for you. It is a well known easy and important fact that: the composite of two fibrations is a fibration. I am ready to pay two bottles of GOOD champagne to anyone who can state this result using only indexed categories, and SIX bottles to anyone who can state, and prove, the same result > > Dear Jean, > >> 1.2.5 Terminology again. There is by now an unwritten but >> (almost) =20 >> unanimous "linguistic consensus" on the following terminology: If >> P =20 >> is a "property" of categories, a category satisfies P locally iff >> all =20= >> >> its slices satisfy P. > > Unfortunately, I was given to understand that there was a different > consensus: that "locally P" means the homsets satisfy P. > > So "locally small" means "with small homsets". and "locally > ordered" means "Poset-enriched". > > I have also heard it said that "V-enriched" was once upon a time > called "locally V-internal". > > For several years I have been writing "locally C-indexed" to mean > "enriched in [C^op,Set]". Equivalently, a locally C-indexed > category D is a strictly C-indexed category where all the fibres > have the same objects ob D, and all the reindexing functors are > identity-on-objects. > > Given that you dislike indexed categories for some reason that you > do not specify (is it only *strict* indexed categories that you > object to?) this usage will probably horrify you... > > >> I quote him again: >> >> "My footnote refers to "other authors" who said that LCCCs should =20 >> have binary products; I think I may have had Thomas Streicher in =20 >> mind, but I don't recall what he may have said or in what paper." >> >> 3.3.2- Streicher &... others? >> Why mention Thomas Streicher without at least trying to find out >> what =20= >> >> he said or wrote precisely on the question? >> Why not mention P. Johnstone's "Elephant" where this is precisely >> =20 >> written, long after Taylor's world famous "footnote" was >> published. =20 >> Lack of courage? Fear for future promotions? > > That's unlikely. Paul Taylor generally says what he thinks to > everyone. I imagine that, when he wrote the footnote, he'd just > read some paper of Thomas Streicher that irked him for some reason. > > BTW, contrary to some of your correspondents, I would argue that > modelling dependent type theory requires a lccct (with extensive > coproducts) rather than a lccc. That is because the contexts of > the type theory are introduced by two rules: empty context and > context extension. If you don't have a terminal object to model > the empty context, surely you don't have a model of dependent type > theory. > > regards > Paul >