From rrosebru@mta.ca Fri Dec 2 07:27:40 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 02 Dec 2005 07:27:40 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Ei90x-0000BA-6q for categories-list@mta.ca; Fri, 02 Dec 2005 07:23:43 -0400 Message-Id: <200512012113.jB1LD6u26795@math-cl-n03.ucr.edu> Subject: categories: Re: higher gauge theory To: categories@mta.ca (categories) Date: Thu, 1 Dec 2005 13:13:05 -0800 (PST) From: "John Baez" In-Reply-To: <200511290711.jAT7BeT11834@math-cl-n03.ucr.edu> from "John Baez" at Nov 28, 2005 11:11:40 PM X-Mailer: ELM [version 2.5 PL6] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit X-Spam-Checker-Version: SpamAssassin 3.0.4 (2005-06-05) on mx1.mta.ca X-Spam-Level: X-Spam-Status: No, score=0.0 required=5.0 tests=none autolearn=disabled version=3.0.4 Sender: cat-dist@mta.ca Precedence: bulk X-IMAPbase: 1144781908 49 Status: O X-Status: X-Keywords: X-UID: 1 Dear Categorists - > Here's a new paper: > > http://math.ucr.edu/home/baez/higher.pdf > http://math.ucr.edu/home/baez/higher.ps > > John Baez and Urs Schreiber > Higher Gauge Theory ... but if that's too long, you can see the results in distilled form here: http://math.ucr.edu/home/baez/union/ This is my talk at this weekend's Union College conference on categories, topology and commutative algebra. Hope to see some of you there! Best, jb From rrosebru@mta.ca Fri Dec 2 07:33:02 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 02 Dec 2005 07:33:02 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Ei998-0000O6-J9 for categories-list@mta.ca; Fri, 02 Dec 2005 07:32:10 -0400 Message-ID: <438F8AEE.9010006@andrej.com> Date: Fri, 02 Dec 2005 00:44:46 +0100 From: Andrej Bauer Reply-To: Andrej.Bauer@andrej.com User-Agent: Debian Thunderbird 1.0.2 (X11/20050602) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Categories References: <546F06B4-60F8-11DA-9B52-000393B90F2C@wanadoo.fr> In-Reply-To: <546F06B4-60F8-11DA-9B52-000393B90F2C@wanadoo.fr> Content-Type: text/plain; charset=ISO-8859-2 Content-Transfer-Encoding: 7bit Subject: categories: Re: Fwd: Mathematica and CAS Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 2 [note from moderator: the poster is right, this discussion is not relevant to this list, and this final post is allowed as a response to the previous post only.] I feel guilty about prolonging a discussion which is not related to categories. If the moderator thinks we should take it off line, we can do so (see the end of this posting). But since the last message was public and it asked for a reaction, I will reply in the same fashion. Thank you to Jean Benabou for posting the message by Jacqueline Zizi. I would like to make a couple of remarks. First of all, I want it to be clear that Jacqueline and I are on the same side. I am a big fan of Computer Algebra Systems (CAS). I promote them at our department and I teach my students how to use them to solve problems they meet in analysis and algebra. I show them examples from their analysis class which their teacher solved incorrectly but Mathematica gets them right. I shall now also teach them how to compute the limit correctly, thanks to Jacqueline's suggestion. By the way, there is no need to compute the power expansion of numerator and denominator separately. You can just compute Series[...] of the whole thing: In[42] := Series[((1 + 4*x^2)^(1/4) - (1 + 5*x^2)^(1/5))/ (a^(-x^2/2) - Cos[x]), {x, 0, 4}] // Simplify Out[42] = x^2/(1 - Log[a]) + (...) x^4 + O[x^5] You see immediately that things go wrong when Log[a] = 1 ... >> But I have the impression that Andrej himself falls in the trap. I fail to see how I fall in _the_ trap, i.e., that I make the same sort of mistakes as students and CAS. I surely hope I do not :-) Perhaps, it was meant that I just made a mistake when I stated: >> And especially when he says that : >> " I guess I am trying to point out that current Computer Alegbra >> Systems are very tricky to use_correctly" This is an observation about user experience, namely that one cannot trust CAS 100% without having a lot of expert knowledge about it. You may have a different experience, but mine is as stated. I find most of Jacqueline's comments to be explanations about why CAS are not 100% mathematically correct all the time: they are complex, they grow with time, etc. This is all true and well, and I am NOT saying that any CAS which is not 100% correct all the time should be eliminated from the face of the Earth. However, I do take issue with the fact that CAS are presented in a dishonest way. A typical CAS demo does not show you that things can go wrong. The documentation does NOT state clearly the conditions under which builtin functions may be used, contrary to what Jacqueline says: >> The rules for application of the primitives are >> clearly given in Mathematica, in the "Help" menu. You make it sound as if a user who reads the help menu for "Limit" will know that free parameters may sometimes lead to errors. Here is the complete text of help: -------- Limit Limit[expr, x->x_0] finds the limiting value of expr when x approaches x_0. * Example: Limit[Sin[x]/x, x->0] --> 1. * Limit[expr, x->x_0, Direction -> 1] computes the limit as x approaches x_0 from smaller values. * Limit[expr, x->x_0, Direction -> -1] computes the limit as x approaches x_0 from larger values. * Limit returns Interval objects to represent ranges of possible values, for example at essential singularities. * Limit returns unevaluated when it encounters functions about which it has no specific information. Limit therefore makes no explicit assumptions about symbolic functions. * See The Mathematica Book: Section 1.5.10 and Section 3.6.8. * See also: Series, Residue. * Related package: NumericalMath`NLimit`. --------- I let the readers judge whether "Limit therefore makes no explicit assumptions about symbolic functions" makes it clear that using parameters in limits can cause wrong answers. Additionally, there is no hint whatsoever that Mathematica will apply l'Hospital rule (or something equivalent) without checking the side condition for it. The reason why conditions of correct usage are not stated clearly is simple: nobody knows them. CAS are so complex and so liberal about which rule gets applied when that it is next to impossible to write down precisely when they will work correctly. I understand that CAS are complex and that we would get nowhere if we worried about correctness all the time. But the makers of CAS should be honest about this: CAS do NOT state clearly the conditions under which they work correctly, therefore it is difficult to know whether they have given a correct answer. Jacqueline herself wonders about correctness of the results she gets in her third example at http://homepage.mac.com/jacquelinezizi/CategoriesQA/. It is an open problem, as far I can tell, to create a powerful CAS with perfect control of correctness. I suggest that we take further discussion off the categories list. One possible forum is my blog at http://math.andrej.com, where I posted further examples of how Mathematica gets things wrong at http://math.andrej.com/2005/12/02/design-of-computer-algebra-systems/ Andrej Bauer From rrosebru@mta.ca Fri Dec 2 14:02:54 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 02 Dec 2005 14:02:54 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EiFEB-0006RG-IK for categories-list@mta.ca; Fri, 02 Dec 2005 14:01:47 -0400 Message-ID: <43906B25.9020802@math.upenn.edu> Date: Fri, 02 Dec 2005 10:41:25 -0500 From: jim stasheff User-Agent: Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.7.2) Gecko/20040804 Netscape/7.2 (ax) X-Accept-Language: en-us, en MIME-Version: 1.0 To: Categories List Subject: categories: nerves Content-Type: text/plain; charset=us-ascii; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 3 Does anyone know a reference for grothedieck's characterization for when a simplicial set is the nerve of a groupoid or if anyone observed it earlier? jim From rrosebru@mta.ca Fri Dec 2 14:02:54 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 02 Dec 2005 14:02:54 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EiFCS-0006IF-W4 for categories-list@mta.ca; Fri, 02 Dec 2005 14:00:01 -0400 X-ME-UUID: 20051202122454739.B4669280019F@mwinf1008.wanadoo.fr From: Philippe Gaucher Reply-To: gaucher@pps.jussieu.fr To: categories@mta.ca Subject: categories: Re: semi-categories Date: Fri, 2 Dec 2005 13:25:46 +0100 User-Agent: KMail/1.7.1 References: <1133389475.438e26a388e62@mail2.buffalo.edu> In-Reply-To: <1133389475.438e26a388e62@mail2.buffalo.edu> MIME-Version: 1.0 Content-Type: text/plain; charset="windows-1252" Content-Disposition: inline Message-Id: <200512021325.46133.gaucher@pps.jussieu.fr> Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 4 Le mercredi 30 Novembre 2005 23:24, vous avez =E9crit : > Perhaps it has not been sufficiently emphasized that semi-categories an= d > the like are not really "generalizations" of categories (though formall= y > they may appear so).=20 Indeed, at least in my case, a flow must not be viewed as a generalizatio= n of=20 the notion of small categories. Let me explain a little bit what I am doi= ng=20 with these objects. I was not very explicit in my previous post. And so t= he=20 terminology I use is not in "competition". I want to model HDA, at least those coming from precubical sets. I use a = set=20 of states X^0 and between each state A and B of the HDA, there is a=20 topological space P_{A,B}X whose elements represent the non-constant=20 execution paths from A to B. The topology of this space models the=20 concurrency of the situation between A and B. And execution paths can be=20 composed with a strictly asssociative law. There does not necessarily exi= st a=20 loop from a given state A to itself : so P_{A,A}X can be empty. This fact= is=20 one reason among several other ones why I remove the identity maps. Inside this model, I am able to define what is a dihomotopy equivalence. = The=20 main problem to define dihomotopy is that some contractible parts of "the= =20 directed spaces of execution paths" must not be contracted. Otherwise in = the=20 categorical localization, the relevant geometric information is lost. In=20 particular, initial and final states must be unchanged by a dihomotopy=20 equivalence. A very simple example : take two execution paths going from = one=20 initial state 0 to one final state 1. If contractions in the direction of= time=20 are allowed, one finds in the same equivalence class a loop. Some example= s of=20 unwanted final states are deadlocks of concurrent systems : a deadlock is= =20 nothing else but a final state from a geometric viewpoint. Flows allow to= =20 propose a solution of this problem : in fact I introduced this notion of=20 flows on purpose, to make the following solution work.=20 The first kind of dihomotopy equivalence is a morphism f:X->Y such that=20 f^0:X^0->Y^0 is a bijection and such that Pf:PX->PY is a weak homotopy=20 equivalence. It turns out that there is a model structure on Flows whose = weak=20 equivalences are exactly the preceding kind of morphisms. By imposing the= =20 condition f^0:X^0->Y^0 bijective, we do not take any risk : nothing is=20 contracted in the direction of time. So no geometric information is lost.= But=20 this kind of identification is too rigide ! The second kind of dihomotopy= =20 equivalence is generated by taking a representative set of inclusions of=20 posets P1\subset P2, where P1 and P2 are finite bounded posets and where = the=20 inclusions preserve the bottom element and the top element (which are=20 different by hypothesis in a bounded poset). For example, the inclusion o= f=20 posets {0<1}\subset{0 Envelope-to: categories-list@mta.ca Delivery-date: Fri, 02 Dec 2005 14:02:54 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EiFDV-0006NY-Qm for categories-list@mta.ca; Fri, 02 Dec 2005 14:01:05 -0400 In-Reply-To: References: Mime-Version: 1.0 (Apple Message framework v733) Content-Type: multipart/alternative; boundary=Apple-Mail-5-605476314 Message-Id: <15DD9EBD-B91E-4386-A143-DD3A4348F0DB@dima.unige.it> From: Marco Grandis Subject: categories: Re: Name for a concept Date: Fri, 2 Dec 2005 14:51:38 +0100 To: categories@mta.ca Content-Transfer-Encoding: quoted-printable Content-Type: text/plain;charset=ISO-8859-1;delsp=yes;format=flowed Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 5 I think that such squares should be called "exact" or =20 "semicartesian" (where cartesian square =3D pb). They should be viewed as the natural self-dual generalisation of =20 pullback and pushout. They appear whenever one studies categories of =20 relations. 1. In an abelian category, I would prefer "exact", or "Hilton-exact". Hilton considered such squares (for abelian categories), and proved =20 that an equivalent condition is that this square (of proper =20 morphisms) is "bicommutative" in the category of relations (i.e. it =20 commutes and stays commutative when you reverse two "parallel" arrows =20= - as relations). Plainly: bicartesian square =3D> pullback =3D> exact; and dually. REFERENCE: P. Hilton, Correspondences and exact squares, in: Proc. Conf. on =20 Categorical Algebra, La Jolla 1965, Springer, pp. 254-271. 2. Studying more general categories of relations, I considered =20 "semicartesian squares" (f,g, h,k), defined - in any category - by =20 the following self-dual property (after being commutative, of course): Whenever (f',g', h,k) and (f,g, h',k') commute, also the =20 external square (f',g', h',k') commutes B f' f h h' A' A D D' g' g k k' C (add slanting arrows f': A' --> B, f: A --> B, etc). - Again: bicartesian square =3D> pullback =3D> semicartesian, and =20= dually. - If pb's and/or po's exist, one can give a lot of equivalent =20 properties; eg: -- (f,g) and the pb of (h,k) have the same po (or the same =20 commutative squares out of them). - In an abelian category, semicartesian amounts to the previous notion. - In Set, it characterises again those squares which are =20 bicommutative in Rel. REFERENCE: M. Grandis, Sym=E9trisations de categories et factorisations =20 quaternaires, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. =20 14 sez. 1 (1977), 133-207. 3. A 2-dimensional version of this property (actually a STRUCTURE on =20 2-cells), was introduced by Guitart, and called "H-exact", if I =20 remember well (H for Hilton) REFERENCES: - R. Guitart, Carr=E9s exacts et carr=E9s deductifs, Diagrammes 6 = (1981), =20 G1-G17. - R. Guitart and L. Van den Bril, Calcul des satellites et =20 pr=E9sentations des bimodules =E0 l'aide des carr=E9s exacts, Cahiers =20= Topologie G=E9om. Diff=E9rentielle 24 (1983), no. 3, 299-330. (and some other papers by the same authors). Best regards Marco Grandis --Apple-Mail-5-605476314 Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset=ISO-8859-1
I think that such squares = should be called "exact" or "semicartesian" (where cartesian square =3D = pb).
They should be viewed as the=A0natural self-dual = generalisation of pullback and pushout. They appear whenever one studies = categories of relations.

1.= In an abelian category, I would prefer "exact", or = "Hilton-exact".

Hilton considered such = squares (for abelian categories), and proved that an equivalent = condition is that this square (of proper morphisms) is "bicommutative" = in the category of relations (i.e. it commutes and stays commutative = when you reverse two "parallel" arrows - as relations).

Plainly: =A0 bicartesian square=A0 =3D>=A0 pullback=A0 =3D>=A0 exact;=A0 and = dually.

REFERENCE:
P. Hilton, Correspondences and = exact squares, in: Proc. Conf. on Categorical Algebra, La Jolla = 1965, Springer, pp. 254-271.

2.= Studying more general categories of relations, I considered = "semicartesian squares"=A0 (f,g, h,k),=A0 defined - in any category - by = the following self-dual property (after being commutative, of = course):

=A0Whenever=A0 (f',g', h,k)=A0 and=A0 (f,g, h',k')=A0 commute, also the external square=A0 (f',g', h',k')=A0 commutes

=A0=A0 =A0=A0 =A0=A0 =A0=A0 = =A0=A0 =A0=A0 =A0=A0 =A0=A0 =A0=A0 =A0B =A0
=A0= =A0 =A0=A0 =A0=A0 =A0f'=A0 =A0=A0 =A0=A0 =A0=A0 =A0f=A0 =A0 =A0=A0 = =A0h = =A0 =A0=A0 =A0=A0= =A0h'
=A0 A'=A0=A0 =A0=A0 =A0=A0 =A0=A0 = =A0A = =A0 =A0=A0 =A0=A0= =A0=A0 =A0=A0 =A0D =A0 =A0=A0 =A0=A0 =A0=A0 =A0D'
=A0= =A0 =A0=A0 =A0=A0 =A0g'=A0=A0 =A0=A0 =A0=A0 =A0g=A0=A0 =A0=A0 = =A0k = =A0=A0 =A0=A0 = =A0=A0 =A0k'
=A0 =A0=A0 =A0=A0 =A0=A0 = =A0=A0 =A0=A0 =A0=A0 =A0=A0 =A0=A0 =A0C

(add slanting arrows=A0 f': A' --> B, =A0f: A = --> B,=A0 =A0etc).

- Again: bicartesian = square=A0 = =3D>=A0 pullback=A0 =3D>=A0 semicartesian,=A0 and dually.

- If pb's=A0 and/or=A0 = po's exist, one can give a lot of equivalent properties; = eg:

--=A0 (f,g)=A0 and=A0the pb of=A0 (h,k) have the same = po (or the same commutative squares out of them).

- = In an abelian category, semicartesian amounts to the previous = notion.
- In Set, it characterises again those squares which = are bicommutative in Rel.

REFERENCE:
M. Grandis, Sym=E9trisations de = categories et factorisations quaternaires, Atti Accad. Naz. Lincei = Mem. Cl. Sci. Fis. Mat. Natur. 14 sez. 1 (1977), 133-207.

3. A 2-dimensional version = of this property (actually a STRUCTURE on 2-cells), was introduced by = Guitart, and called "H-exact", if I remember well (H for = Hilton)

REFERENCES:

- R. Guitart, Carr=E9s = exacts et carr=E9s deductifs, Diagrammes 6 (1981), = G1-G17.
- R. Guitart and L. Van den Bril, Calcul des = satellites et pr=E9sentations des bimodules =E0 l'aide des carr=E9s = exacts, Cahiers Topologie G=E9om. Diff=E9rentielle 24 (1983), no. 3, = 299-330.
(and some other papers by the same = authors).

Best regards

Marco = Grandis

= --Apple-Mail-5-605476314-- From rrosebru@mta.ca Fri Dec 2 14:02:54 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 02 Dec 2005 14:02:54 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EiFBf-0006Ey-Ve for categories-list@mta.ca; Fri, 02 Dec 2005 13:59:12 -0400 Message-ID: <003801c5f732$57cff880$06fb4c51@brown1> From: "Ronald Brown" To: "Categories list" References: Subject: categories: Re: Name for a concept Date: Fri, 2 Dec 2005 11:19:22 -0000 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: 6 This kind of condition occurs in topology as a fibrant square - but where all the maps are fibrations as is the map A ---> B x_D C . This can be generalised to cubes. See a paper by R. Steiner on `Resolutions of spaces by n-cubes of fibrations', J. London Math. Soc.(2), 34, 169-176, 1986 used to build a complete (strict) algebraic model of homotopy n-types which allows some computations. This raises the spectre in algebra of Resolutions of A by free crossed n-cubes of A. to give a more `nonabelian' homological algebra. Of course crossed n-cubes of A should be equivalent to n-fold groupoids in A. This would presumably bring in higher versions of nonabelian tensor products in A; a bibliography of such a tensor, mainly for n=2, has 90 items. This probably does not help to answer Mike's question on the name! Ronnie www.bangor.ac.uk/r.brown/nonabtens.html ----- Original Message ----- From: "Michael Barr" To: "Categories list" Sent: Thursday, December 01, 2005 1:48 AM Subject: categories: Name for a concept > Is there a standard name for a square > A ----> B > | | > | | > | | > v v > C ----> D > in which the canonical map A ---> B x_D C is epic? I had always called it > a weak pullback, but Peter Freyd claims that that phrase is reserved for > the case that it satisfies the existence, but not necessarily the > uniqueness of the definition of pullback. In fact, he claims it means > that Hom(E,-) converts it to the kind of square I am talking about. > What is interesting is that in an abelian category, it satisfies > this condition iff it satisfies the dual condition iff the evident > sequence A ---> B x C ---> D is exact. Putting a zero at the left end > characterizes a genuine pullback and at the other end a pushout. > > Michael > > > From rrosebru@mta.ca Mon Dec 5 16:07:06 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 05 Dec 2005 16:07:06 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EjMSo-0000Ue-S9 for categories-list@mta.ca; Mon, 05 Dec 2005 15:57:30 -0400 Date: Mon, 5 Dec 2005 14:10:00 +0000 (GMT) From: "Prof. Peter Johnstone" To: Categories mailing list Subject: categories: Chalkfinger Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 7 Peter Freyd's recent posting of the `Categories Anonymous' skit prompted me to dig out and transcribe an even older skit which originated with the Chicago Junior Math Club, and which stars Saunders Mac Lane as the evil villain Chalkfinger. I don't know its exact date: it should be datable by the reference at the end to Chalkfinger's visit to Japan, but although Saunders refers to that visit in his autobiography (p. 297) he doesn't quote the date. The typescript copy which I have (and which I acquired from Murray Adelman in 1984) was apparently revised for performance at the Bowdoin Summer Advanced Seminar in Homological Algebra (again, I don't know the year). If anyone knows more details of the history of the skit, I'd be interested to hear them. Peter Johnstone ++++++++++++++++++++ DRAMATIS PERSONAE: K.: Kaplansky Chestcough: Ann Chertkoff (secretary) Bailout: Baily Bond: Herstein Chalkfinger: Mac Lane Manishevitz: Liulevicius Filterfish: Amitsur Oddprime: Applied Math (?) Pedro: Calderon Ziggy: Zygmund Browser (Felix) Browder SCENE 1 Opening: [Goldfinger theme heard. Jacobson writes on blackboard] Jacobson: Good evening, welcome to the Junior Math Club's presentation of Chalkfinger. [He is dragged off. A scream is heard offstage,] [Desk of K, chair for Miss Chestcough] K: Miss Chestcough! Chest: [cough] Yes? [Bailout runs in.] Bailout: Who coughed? Chest: Me, Professor Bailout. Bailout: Well, don't let it happen again. [exit] Chest: {aside] That was a young right idealist. K: Miss Chestcough, see if you can get Agent 00\pi on the red telephone. Chest: At 8.30 in the morning? Well, I'll try K., but 00\pi is always so irrational at this hour. [exits to call] K: Hmmm. How does she know that? [Chestcough returns, after a minute Bond enters] Bond: [flirting] Hello, pigeon. Why do I chase around the world when there are girls like you at home? Chest: [smiles] You'd better go in to see K., but I'll be here when you get back. [Bond crosses to K.'s desk, sits down. K. stares at him for *long* moment.] K: Could you give me an epsilon of your time? Bond: Make it epsilon over 2. What's up, Boss? K: Well, it's Chalkfinger. He's been at our chalk supply again. Why, that man is responsible for most of the world's illicit traffic in colored chalk. Bond: But what does he use it for? K: You won't believe this. He proves theorems about categories! Bond: [laughs] That's ridiculous! [does double-take] What's a category? K: Why, don't you know? You're not up with the times, Bond. As I understand it, categories arose from Pavlov's experiments with dogs. Pavlov trained dogs to expect meat at the sound of a bell. Then he took the meat away, but the dogs continued to salivate when the bell was rung. Now Chalkfinger has taken the meat out of mathematics, but mathematicians still salivate at the sound of the bell. Bond: Insidious! K: You bet. Now I want you to infiltrate their organization. Bond: But suppose I don't make the grade? K: You'll do it, Yitz. I'm putting my money on Israel Bond. Bond: Thank you, sir. I'll try to protect your interest. Say, whatever happened to that last case you were working on? K: I couldn't get anywhere on it, so I turned it over to a graduate student. Now, for your equipment: [they stand up] We've prepared for you a white Jaguar complete with left and right annihilators, deformation retract and portable pool table. Here's the key to the locked stall in the third floor john. We'll communicate as usual by leaving messages under the seat. Use all the paper you want, and good luck! [Exeunt -- Alperin and Thompson make first crossing of stage] SCENE 2 [Students playing chess] Student 1: Going to the colloquium today? All the big operators from Spectrum are going to be there: Big Ziggy, Big Pedro and even Big Browser, their [gestures] elliptic operator. Student 2: Nah, I'm too busy playing chess. Student 1: Oh, Chalkfinger won't like that. [long pause -- Chalkfinger enters with Oddprime] Chalk: By thunder, you know what can happen to students like you? Oddprime! Hat! [Oddprime throws frisbee, something breaks] Chalk: Hah, Oddprime can divide anything. [Bond enters] Welcome to our organization, Mr Bond. We're looking forward to your course in Hopf algebras. Let me show you around. Manishevitz, here, is working on a computer program to determine the first odd square. Manish: My name is Manishevitz, but you can call me Manishevitz if you want. Bond: That looks like difficult work. Manish: Any child in first grade could do it. It's all a matter of gamesmanship. Chalk: Good man to talk to, Bond, if you can get through the queue at his door. Over there is Filterfish, still trying to disprove that the Arabs invented algebra. Filter: [whispers] What is the password? Bond: Ultra-filter. Filter: You must be the man from K. Bond: Your image is my kernel, Daddy-O. Filter: Exactly. Chalk: Mr Bond, do you subscribe to the Dieudonne doctrine or the category of vector spaces? Bond: Category, shmategory, I don't go for that crap. Chalk: Categories crap? The man's a spy! Seize him! [all seize him] What do we do with spies? Manish: We could boil him to death in Eckhart 312. Filter: Or make him teach Math 101. Chalk: I know, we'll put him trough the pull-back. Oddprime: Heh, heh, heh, through the push-out and the pull-back! All: Through the pull-back! [They drag Bond out] Bond: You may torture my body, but you'll never annihilate my ideals. [Screams heard offstage, they bring back his unconscious body and throw it on the floor. Oddprime stands guard. Pussy enters] Pussy: Izzy dead? Oddprime: No, Izzy just tired. [Bond rises slowly, shakes himself] Bond: Well, hello. Pussy: Hi! My name is Pussy Galois. Bond: [half-fainting again] Wow, are you ever projective! Pussy: Not only that, I'm free. Bond: [stands up] I'd like to prove that. Pussy: It's been tried before but [sitting down] maybe your techniques are better. Bond: [joining her] Say, Pussy, how does Chalkfinger smuggle his chalk out of the building? Pussy: Oh, haven't you ever figured that out? He hides it in his baggy pants. Bond: Ingenious! [Alperin and Thompson make second crossing] Bond: What in the world? Pussy: Oh, they always do that. Come with me: [takes his hand] we'll eavesdrop on Chalkfinger's meeting with Spectrum. SCENE 3 [Ziggy, Pedro and Browser on stage] Pedro: What'cha got there, Big Ziggy? Ziggy: It's Chalkfinger's latest masterpiece, which I've acquired at considerable expense Pedro: Let's see: [reads] 'In the beginning Chalkfinger created the category and the functor, but the category was void and the functor was forgetful. And Chalkfinger said "Let there be maps", and there were maps. And Chalkfinger called the maps morphisms, and the members of the category he called objects, and Chalkfinger thought that was pretty good, and so did his friend Sammy. And there was evening and there was morning in 1945.' Ziggy: This beats analysis. Browser: Sounds like a very theological subject. [takes the book] 'And on the sixth day, Chalkfinger created mathematicians and set them to work in the categories' -- oh, surely that's going too far! Pedro: Shh! Here he comes. [Chalkfinger and Lin Ton enter] Chalk: Gentlemen, meet my Chinese assistant Lin Ton. I never understand you analysts, so I've brought along Lin Ton to interpret for me. Lin Ton: Have you brought the shipment? Pedro: Yes, it's all here. [They deposit chalk in a box] We had some trouble, though. Bailout keeps destroying the red chalk. Ziggy: I notice you keep the colored chalk in a separate box. Lin Ton: Separate but equal. Browser: Maybe we can integrate it while we're here. Chalk: You know, I had the strangest dream last night. I dreamed that I was surrounded by seven lean sheaves and seven flabby sheaves, and the seven lean sheaves came and devoured the seven flabby sheaves. Pedro: A singular dream! Ziggy: I think you need a different kind of analyst. Browser: Yeah, now look, Chalkfinger. We've all made our deliveries of colored chalk. You promised us that your categories would yield us theorems. This is Ext-Tor-tion! Chalk: Gentlemen, I wouldn't dream of going back on a promise. If you want, I'll give you a theorem apiece; but first I will demonstrate my new secret weapon. Pedro: Is he on the level? Ziggy: I don't know. It's hard to take the measure of the man. Chalk: You are about to witness the power of the Grothendieck ring! [Pulls out ring] Lin Ton: That ring is nil, try this one. Browser: With that ring, I thee dread. Chalk: You'd better ... [pushing him away] because none of you are getting out of here alive! [He runs offstage] [Hissing noises heard from left and right] Lin Ton: Oh no, it's the thermostats! Ziggy: Poison gas. Browser: [on left] Ahh, I'm being annihilated on the left! Pedro: [on right] And I on the right! [All collapse; Chalkfinger appears in front] Chalk: [aside] Hah, a swift death for a Swift Professor. Bond: I saw the whole thing, Chalkfinger. Your heinous crimes will not go unpunished. [Chalkfinger trips Bond, they fall, the ring slides away] Pussy: The Grothendieck ring -- it's going to explode! [Explosion noise, chairs overturned, everybody collapses] [Alperin and Thompson make final crossing, oblivious to everything] [Bond gets up, brushes himself off] Bond: Good thing I'm immune to radical rings! And so the tragic tale fo Chalkfinger comes to its untimely end. The explosion of the Grothendieck ring was powerful enough to cast Chalkfinger all the way to Japan. Because of this affair, K. had to hide out in England for a year, and I've earned an extended vacation on the Mediterranean. FINIS From rrosebru@mta.ca Mon Dec 5 16:07:06 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 05 Dec 2005 16:07:06 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EjMUc-0000cY-8W for categories-list@mta.ca; Mon, 05 Dec 2005 15:59:22 -0400 Subject: categories: Re: Name for a concept From: Eduardo Dubuc To: categories@mta.ca (Categories list) Date: Mon, 5 Dec 2005 13:16:13 -0300 (ART) In-Reply-To: from "Michael Barr" at Nov 30, 2005 08:48:52 PM X-Mailer: ELM [version 2.5 PL2] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-Id: X-Spam-Checker-Version: SpamAssassin 3.0.4 (2005-06-05) on mx1.mta.ca X-Spam-Level: X-Spam-Status: No, score=0.0 required=5.0 tests=none autolearn=disabled version=3.0.4 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 8 > > Is there a standard name for a square > A ----> B > | | > | | > | | > v v > C ----> D > in which the canonical map A ---> B x_D C is epic? These are called "quasi-pullbacks" by Joyal, and they form a class of "open maps" in the category of squares. The pullbacks form the corresponding class of etal maps. These two classes are essential for the development of the theory (etal class and open class in the sense of Joyal). There are published articles by Joyal and Moerdijk on the subject. From rrosebru@mta.ca Tue Dec 6 19:53:52 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 06 Dec 2005 19:53:52 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EjmV4-0000KP-UI for categories-list@mta.ca; Tue, 06 Dec 2005 19:45:34 -0400 Mime-Version: 1.0 (Apple Message framework v733) Content-Type: text/plain; charset=ISO-8859-1; delsp=yes; format=flowed Message-Id: <41D4B4D6-BBF1-4371-9339-046EA2ADACF2@dima.unige.it> Content-Transfer-Encoding: quoted-printable From: Marco Grandis Subject: categories: Re: Name for a concept Date: Mon, 5 Dec 2005 15:44:26 +0100 To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 9 In reply to M. Barr's posting. Richard Wood tells me that my posting on this subject, dated 2 =20 December 2005, was unreadable with 'elm' and nearly so with 'pine', =20 due to rich-text marks. I am reposting it in plain text (hopefully), with a few small =20 additions - and apologies MG _____ I think that such squares should be called "exact" or =20 "semicartesian" (where cartesian square =3D pb, cocartesian =3D po). They should be viewed as the natural self-dual generalisation of =20 pullback and pushout (and their name should be "self-dual", in some =20 way). They appear whenever one studies categories of relations. 1. In an abelian category (where they are chracterised by the exact =20 sequence you have mentioned), I would prefer "exact", or "Hilton-exact". Hilton considered such squares (for abelian categories), and proved =20 that an equivalent condition is that this square (of proper =20 morphisms) is "bicommutative" in the category of relations (i.e. it =20 commutes and stays commutative when you reverse two "parallel" arrows =20= - as relations). Plainly: bicartesian square =3D> pullback =3D> exact; and dually. REFERENCE: P. Hilton, Correspondences and exact squares, in: Proc. Conf. on =20 Categorical Algebra, La Jolla 1965, Springer, pp. 254-271. 2. Studying more general categories of relations, I considered =20 "semicartesian squares" (f,g, h,k), defined - in any category - as =20 the commutative squares satisfying the following self-dual property: Whenever (f',g', h,k) and (f,g, h',k') commute, also the outer =20 square (f',g', h',k') commutes B f' f h h' A' A D D' g' g k k' C (add slanting arrows f': A' --> B, g': A --> C, f: A --> B, etc). - Again: bicartesian square =3D> pullback =3D> semicartesian, and =20= dually. - If pb's and/or po's exist, there are a lot of equivalent =20 properties; eg: -- (f,g) and the pb of (h,k) have the same po (or the same =20 commutative squares out of them). - In an abelian category, semicartesian amounts to the previous notion. - In Set, it characterises again those squares which are =20 bicommutative in Rel. REFERENCE: M. Grandis, Sym=E9trisations de categories et factorisations =20 quaternaires, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. =20 14 sez. 1 (1977), 133-207. 3. A 2-dimensional version of this property (actually a STRUCTURE on =20 2-cells), was introduced by Guitart, and called "H-exact", if I =20 remember well (H for Hilton) REFERENCES: - R. Guitart, Carr=E9s exacts et carr=E9s deductifs, Diagrammes 6 = (1981), =20 G1-G17. - R. Guitart and L. Van den Bril, Calcul des satellites et =20 pr=E9sentations des bimodules =E0 l'aide des carr=E9s exacts, Cahiers =20= Topologie G=E9om. Diff=E9rentielle 24 (1983), no. 3, 299-330. (and some other papers by the same authors). Best regards Marco Grandis From rrosebru@mta.ca Tue Dec 6 19:53:52 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 06 Dec 2005 19:53:52 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EjmZR-0000Zp-9K for categories-list@mta.ca; Tue, 06 Dec 2005 19:50:05 -0400 Date: Tue, 6 Dec 2005 11:12:54 +0100 Mime-Version: 1.0 (Apple Message framework v543) Content-Type: text/plain; charset=ISO-8859-1; format=flowed Subject: categories: Re: Name for a concept From: jean benabou To: Categories Content-Transfer-Encoding: quoted-printable Message-Id: Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 10 Continuation of namings (1) - Is there a standard name for the squares where the canonical map=20= is monic , i.e. the pair of maps A --->B and A --->C is jointly =20 monic. I propose semi-pullback (2)- In most cases the canonical map being epic is not what one really=20= wants. Of course Joyal assumes the category where the maps live to be a=20= pre-topos, then it's enough, otherwise one cannot "compose" such=20 squares. Do we have to rename the squares where the canonical map is a=20= universal epi, or those where its a universal regular epi? In view of (1), one would like to say that a square is a pullback iff=20 it is both a quasi and semi pullback D=E9but du message r=E9exp=E9di=E9 : > De: Eduardo Dubuc > Date: Lun 5 d=E9c 2005 17:16:13 Europe/Paris > =C0: categories@mta.ca (Categories list) > Objet: categories: Re: Name for a concept > >> >> Is there a standard name for a square >> A ----> B >> | | >> | | >> | | >> v v >> C ----> D >> in which the canonical map A ---> B x_D C is epic? > > > These are called "quasi-pullbacks" by Joyal, and they form a class of > "open maps" in the category of squares. The pullbacks form the > corresponding class of etal maps. These two classes are essential for=20= > the > development of the theory (etal class and open class in the sense of > Joyal). There are published articles by Joyal and Moerdijk on the > subject. > > > > > From rrosebru@mta.ca Tue Dec 6 19:53:53 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 06 Dec 2005 19:53:53 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EjmXL-0000T1-4s for categories-list@mta.ca; Tue, 06 Dec 2005 19:47:55 -0400 Message-ID: <001501c5f9ed$f8ef39c0$6401a8c0@MANESCO> From: "Ernie Manes" To: "Categories mailing list" References: Subject: categories: Re: Chalkfinger Date: Mon, 5 Dec 2005 17:48:12 -0500 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: RO X-Status: X-Keywords: X-UID: 11 I remember the skit well, because it named my thesis advisor-to-be after his somewhat oriental looks as Lin Ton. That puts the skit as the year Fred spent many months at Chicago which was the year before the Bowdoin conference on homological algebra which was held in the summer of 1965 as an NSF summer Advanced Seminar. It was at the seminar that I first met Fred and I did not meet him the previous year precisely because he was at Chicago. Ernie Manes ----- Original Message ----- From: "Prof. Peter Johnstone" To: "Categories mailing list" Sent: Monday, December 05, 2005 9:10 AM Subject: categories: Chalkfinger > Peter Freyd's recent posting of the `Categories Anonymous' skit > prompted me to dig out and transcribe an even older skit which > originated with the Chicago Junior Math Club, and which stars > Saunders Mac Lane as the evil villain Chalkfinger. I don't know its > exact date: it should be datable by the reference at the end to > Chalkfinger's visit to Japan, but although Saunders refers to that > visit in his autobiography (p. 297) he doesn't quote the date. The > typescript copy which I have (and which I acquired from Murray > Adelman in 1984) was apparently revised for performance > at the Bowdoin Summer Advanced Seminar in Homological Algebra (again, > I don't know the year). If anyone knows more details of the history > of the skit, I'd be interested to hear them. > > Peter Johnstone > From rrosebru@mta.ca Tue Dec 6 19:54:38 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 06 Dec 2005 19:54:38 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Ejmd7-0000oS-1S for categories-list@mta.ca; Tue, 06 Dec 2005 19:53:53 -0400 X-ME-UUID: 20051206181221664.A214D1C00142@mwinf3002.me.freeserve.com Message-ID: <006b01c5fa8f$e2502e20$61f94c51@brown1> From: "Ronald Brown" To: Subject: categories: Terminology question wrt fibrations of categories. Date: Tue, 6 Dec 2005 18:06:40 -0000 MIME-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 12 I am writing about matters to do with computation of colimits of a category X in terms of colimits of a category B when there is a bifibration P: X --> B. Terminology already in use is P is cartesian P is cocartesian a lifting of u in B to \phi in X may be cartesian, cocartesian on the other hand Paul Taylor, following Peter Johnstone, I understand, uses \phi is prone, supine, instead of cartesian, cocartesian For the cofibration (?opfibration?) Ob: Groupoids --> Sets, Philip Higgins (1971) and I (1968) have previously used `universal' for cocartesian. In this situation, I would be happier with say 0-final instead of universal. But `supine' does not ring a bell with me, and carries a pejorative tone. Maybe for the general situation P: X --> B we could use P-initial, P-final morphism in X for cartesian, cocartesian morphism which would at least carry some intuition as to the meaning. Comments? I need to make a decision soon for the revision of my old topology book. Not much will be changed, and I might leave the old terminology and refer to more modern uses. However for the book on Nonabelian algebraic topology, I really do need to use modern terminologym, whatever that is, so it would be best to be consistent. I have been looking at Thomas Streicher's notes on fibrations, and at Paul Taylor's Practical Foundations. For my interest, see slides of a recent seminar at Oxford www.bangor.ac.uk/r.brown/oxford2811105.pdf called `Induced constructions and their computation'. Ronnie Brown www.bangor.ac.uk/r.brown From rrosebru@mta.ca Tue Dec 6 19:56:09 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 06 Dec 2005 19:56:09 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Ejmef-0000vI-4p for categories-list@mta.ca; Tue, 06 Dec 2005 19:55:29 -0400 To: categories@mta.ca Subject: categories: CMCS 2006 Date: Tue, 06 Dec 2005 18:47:22 +0000 From: Neil X Ghani Message-ID: <200512061847.aa20447@pat.Cs.Nott.AC.UK> MIME-Version: 1.0 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 13 This is the 2nd Call for Papers for CMCS 2006 CMCS 2006 8th International Workshop on Coalgebraic Methods in Computer Science http://conferences.inf.ed.ac.uk/cmcs06/cmcs06.html Vienna, Austria March 25-27, 2006 The workshop will be held in conjunction with the 9th European Joint Conferences on Theory and Practice of Software ETAPS 2006 March 25 - April 2, 2006 Aims and Scope During the last few years, it has become increasingly clear that a great variety of state-based dynamical systems, like transition systems, automata, process calculi and class-based systems, can be captured uniformly as coalgebras. Coalgebra is developing into a field of its own interest presenting a deep mathematical foundation, a growing field of applications and interactions with various other fields such as reactive and interactive system theory, object oriented and concurrent programming, formal system specification, modal logic, dynamical systems, control systems, category theory, algebra, analysis, etc. The aim of the workshop is to bring together researchers with a common interest in the theory of coalgebras and its applications. The topics of the workshop include, but are not limited to: the theory of coalgebras (including set theoretic and categorical approaches); coalgebras as computational and semantical models (for programming languages, dynamical systems, etc.); coalgebras in (functional, object-oriented, concurrent) programming; coalgebras and data types; (coinductive) definition and proof principles for coalgebras (with bisimulations or invariants); coalgebras and algebras; coalgebraic specification and verification; coalgebras and (modal) logic; coalgebra and control theory (notably of discrete event and hybrid systems). The workshop will provide an opportunity to present recent and ongoing work, to meet colleagues, and to discuss new ideas and future trends. Previous workshops of the same series have been organized in Lisbon, Amsterdam, Berlin, Genova, Grenoble, Warsaw and Barcelona. The proceedings appeared as Electronic Notes in Theoretical Computer Science (ENTCS) Volumes 11,19, 33, 41, 65.1, 82.1 and 106. You can get an idea of the types of papers presented at the meeting by looking at the tables of contents of the ENTCS volumes from those workshops ENTCS Location CMCS 2006 will be held in Vienna on March 25-27, 2006. It will be a satellite workshop of ETAPS 2006, the European Joint Conferences on Theory and Practice of Software. Programme Committee John Power (chair,Edinburgh), Luis Barbosa (Minho), Neil Ghani (Nottingham), H. Peter Gumm (Marburg), Marina Lenisa (Udine), Stefan Milius (Braunschweig), Larry Moss (Bloomington), Jan Rutten (Amsterdam), Hendrik Tews (Dresden), Tarmo Uustalu (Tallinn), Hiroshi Watanabe (Osaka). Keynote Speaker: Peter O'Hearn (Queen Mary, University of London) Invited Speakers: Corina Cirstea (University of Southampton) Alexander Kurz (University of Leicester) Submissions Two sorts of submissions will be possible this year: Papers to be evaluated by the programme committee for inclusion in the ENTCS proceedings: These papers must be written using ENTCS style files and be of length no greater than 20 pages. They must contain original contributions, be clearly written, and include appropriate reference to and comparison with related work. If a submission describes software, software tools, or their use, it should include all source code that is needed to reproduce the results but is not publicly available. If the additional material exceeds 5 MB, URL's of publicly available sites should be provided in the paper. Short contributions: These will not be published but will be compiled into a technical report of the University of Nottingham. They should be no more than two pages and may describe work in progress, summarise work submitted to a conference or workshop elsewhere, or in some other way appeal to the CMCS audience. Both sorts of submission should be submitted in postscript or pdf form as attachments to an email to cmcs06@cs.nott.ac.uk. The email should include the title, corresponding author, and, for the first kind of submission, a text-only one-page abstract. After the workshop, we expect to produce a journal proceedings of extended versions of selected papers to appear in Theoretical Computer Science. Important Dates Deadline for submission of regular papers: January 8, 2006. Notification of acceptance of regular papers: February 6, 2006. Final version for the preliminary proceedings: February 13, 2006. Deadline for submission of short contributions: February 28, 2006. Notification of acceptance of short contributions: March 6, 2006. For more information, please write to cmcs06@cs.nott.ac.uk. _______________________________________________ cmcs06 mailing list cmcs06@cs.nott.ac.uk http://www.cs.nott.ac.uk/mailman/listinfo/cmcs06 This message has been checked for viruses but the contents of an attachment may still contain software viruses, which could damage your computer system: you are advised to perform your own checks. Email communications with the University of Nottingham may be monitored as permitted by UK legislation. From rrosebru@mta.ca Wed Dec 7 20:20:05 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 07 Dec 2005 20:20:05 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Ek9Nn-0001M8-RL for categories-list@mta.ca; Wed, 07 Dec 2005 20:11:35 -0400 Mime-Version: 1.0 (Apple Message framework v619.2) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: <0383b8d97a4e9a4c0fbb6e5d92bf1d5b@pps.jussieu.fr> Content-Transfer-Encoding: 7bit From: Curien Pierre-Louis Subject: categories: goedel prize 2006 (deadline for nominations Jan. 31, 2006) Date: Wed, 7 Dec 2005 09:40:37 +0100 To: categories@mta.ca X-Mailer: Apple Mail (2.619.2) X-Greylist: Sender IP whitelisted, not delayed by milter-greylist-1.7.2 (shiva.jussieu.fr [134.157.0.166]); Wed, 07 Dec 2005 09:35:57 +0100 (CET) X-Antivirus: scanned by sophie at shiva.jussieu.fr X-Miltered: at shiva.jussieu.fr with ID 43969EEA.000 by Joe's j-chkmail (http://j-chkmail.ensmp.fr)! Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 14 I'd like to draw your attention on the call for nominations for GOEDEL Prize 2006 (deadline January 31, 2006) All information on the nomination process can be found on the following url: http://www.eatcs.org/ or http://sigact.acm.org/prizes/godel/ From both sites you can retrieve the pdf file of the 2006 call which has all details about how to submit a nomination. Best regards, On behalf of the Award Committee Pierre-Louis Curien (chair for 2006) From rrosebru@mta.ca Wed Dec 7 20:20:05 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 07 Dec 2005 20:20:05 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Ek9OV-0001Oj-KK for categories-list@mta.ca; Wed, 07 Dec 2005 20:12:19 -0400 Date: Wed, 7 Dec 2005 09:05:10 +0000 (GMT) From: "Prof. Peter Johnstone" To: categories@mta.ca Subject: categories: Re: Terminology question wrt fibrations of categories. In-Reply-To: <006b01c5fa8f$e2502e20$61f94c51@brown1> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 15 On Tue, 6 Dec 2005, Ronald Brown wrote: > on the other hand Paul Taylor, following Peter Johnstone, I understand, uses > \phi is prone, supine, instead of cartesian, cocartesian > `Prone' and `supine' were invented by Paul Taylor; I copied them from him, not the other way round. I'm sorry Ronnie doesn't like them; they seem to me a very neat way of finding two words that both mean `lying horizontally' but have an opposite handedness about them. Peter Johnstone From rrosebru@mta.ca Wed Dec 7 20:20:05 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 07 Dec 2005 20:20:05 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Ek9RD-0001Xa-3a for categories-list@mta.ca; Wed, 07 Dec 2005 20:15:07 -0400 Date: Wed, 7 Dec 2005 08:36:12 -0500 (EST) From: Peter Freyd Message-Id: <200512071336.jB7DaCmS021736@saul.cis.upenn.edu> To: categories@mta.ca Subject: categories: Re: Name for a concept Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 16 Jean asks: Is there a standard name for the squares where the canonical map is monic , i.e. the pair of maps A --->B and A --->C is jointly monic. In the early 60s at the annual AMS meeting held at Denver, Eilenberg, Mac Lane and I sat down to "settle" the terminology. ("Denver One" I called it.) There were just two things we totally agreed on: "weak" is the operator on definitions that removes uniqueness conditions and "partial" the operator that removes existence conditions. So the answer to Jean's question would be "partial pullback". As for the other side -- when the pair of maps are jointly epic -- I've seen them called "near-pullbacks" in the theoretical computer science community. Functors between regular categories that preserve near-pullbacks are precisely those that preserve "weak tabulations" of (n-ary) relations. From rrosebru@mta.ca Wed Dec 7 20:20:06 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 07 Dec 2005 20:20:06 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Ek9PD-0001RE-Ph for categories-list@mta.ca; Wed, 07 Dec 2005 20:13:03 -0400 In-Reply-To: References: Mime-Version: 1.0 (Apple Message framework v733) Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed Message-Id: <22A03972-1301-46FF-BABC-B1CB6C59710E@dima.unige.it> Content-Transfer-Encoding: 7bit From: Marco Grandis Subject: categories: Re: Name for a concept Date: Wed, 7 Dec 2005 12:04:59 +0100 To: categories@mta.ca Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 17 I do not know the original problem of M. Barr, may be he is really interested in getting an epimorphism onto the pullback. However - I apologise for insisting - I think that the important notion for such squares should be a natural self-dual generalisation of pullbacks and pushouts, based on commutative squares and nothing else - so that, in particular, it cannot depend on the variation of monos or epis one is interested in. The one I have proposed in that old paper (under the name of "semicartesian square") is of this kind: - the square (f,g; h,k) commutes, and for every span (f',g') which commutes with the cospan (h,k) and every cospan (h',k') which commutes with the span (f,g), the new span and cospan form a commutative square. All this comes from the obvious Galois connection between sets of spans and cospans (in an arbitrary category), derived from the commutativity relation. Explicitly, let us start with two fixed objects A, B. Let S be the set of spans from A to B: x = (f: C -> A, g: C -> B) (for arbitrary C) and C the set of cospans y = (h: A -> D, k: B -> D) (for arbitrary D). Take their set of parts, PS and PC, ordered by inclusion, and the following (contravariant) Galois connection between them (X in PS, Y in PC): R(X) = set of cospans which commute with all the spans in X, L(Y) = set of spans which commute with all the cospans in Y. Now, a square (x, y) (span/cospan) commutes iff {x} is contained in L({y}) iff {y} is contained in R({x}). A square (x, y) is "semicartesian" (or "exact") iff it satisfies the stronger, equivalent conditions: 1. R{x} = RL({y}) 2. L{y} = LR({x}). Marco Grandis ------------------------- On 1 Dec 2005, at 02:48, Michael Barr wrote: > Is there a standard name for a square > A ----> B > | | > | | > | | > v v > C ----> D > in which the canonical map A ---> B x_D C is epic? I had always > called it > a weak pullback, but Peter Freyd claims that that phrase is > reserved for > the case that it satisfies the existence, but not necessarily the > uniqueness of the definition of pullback. In fact, he claims it means > that Hom(E,-) converts it to the kind of square I am talking about. > What is interesting is that in an abelian category, it satisfies > this condition iff it satisfies the dual condition iff the evident > sequence A ---> B x C ---> D is exact. Putting a zero at the left end > characterizes a genuine pullback and at the other end a pushout. > > Michael > > > > > From rrosebru@mta.ca Wed Dec 7 20:20:06 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 07 Dec 2005 20:20:06 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Ek9Mi-0001El-0W for categories-list@mta.ca; Wed, 07 Dec 2005 20:10:28 -0400 Date: Tue, 6 Dec 2005 16:58:26 -0800 From: Toby Bartels To: Categories Subject: categories: Re: Name for a concept Message-ID: <20051207005826.GA13898@math-rs-n04.ucr.edu> References: Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline In-Reply-To: User-Agent: Mutt/1.4i X-Junkmail-Status: score=0/65, host=sentoku.ucr.edu X-Spam-Checker-Version: SpamAssassin 3.0.4 (2005-06-05) on mx1.mta.ca X-Spam-Level: X-Spam-Status: No, score=0.0 required=5.0 tests=none autolearn=disabled version=3.0.4 Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 18 jean benabou wrote in part: >(1) - Is there a standard name for the squares where the canonical map >is monic , i.e. the pair of maps A --->B and A --->C is jointly >monic. I propose semi-pullback How about "sub-pullback"? since it is a sub-object of the pullback (if there is one). -- Toby From rrosebru@mta.ca Fri Dec 9 09:34:45 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 09 Dec 2005 09:34:45 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EkiJ7-0004JL-Ii for categories-list@mta.ca; Fri, 09 Dec 2005 09:29:05 -0400 Message-ID: <43981850.2080902@math.unice.fr> Date: Thu, 08 Dec 2005 12:26:08 +0100 From: "Clemens.BERGER" Reply-To: cberger@math.unice.fr Organization: Laboratoire J.-A. =?ISO-8859-1?Q?Dieudonn=E9?= User-Agent: Mozilla/5.0 (X11; U; Linux i686; en-US; rv:1.6) Gecko/20040116 X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: name for a concept Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 19 In my previous message, one should read: a commutative square with two parallel fibrations in a right proper model category is a homotopy pullback if and only if the square is exact with respect to the class of weak equivalences. These special squares compose because weak equivalences are stable under base change along fibrations (this is the definition of right proper). Clemens Berger. From rrosebru@mta.ca Fri Dec 9 09:34:45 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 09 Dec 2005 09:34:45 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EkiHG-0003zv-Ha for categories-list@mta.ca; Fri, 09 Dec 2005 09:27:10 -0400 Date: Wed, 07 Dec 2005 23:44:02 -0500 From: Fred E.J.Linton To: "Categories mailing list" Subject: categories: Re: Chalkfinger X-Mailer: USANET web-mailer (CM.0402.7.36) Mime-Version: 1.0 Message-ID: <927JLHeSc4080S07.1134017042@uwdvg007.cms.usa.net> Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 20 Here's my take on this skit, as filtered through 40 years of ever-more-porous memory: Lin Ton was Chalkfinger's "Chinese assistant." I think the Chinese connection came about less because I looked Chinese, as because (a) I was a somewhat mysterious figure, doing no teaching (being a 1-year research associate on one of Saunders' grants), and (b) my name lent itself to Sinification. Others spoofed here, aside from me (Lin Ton) and Saunders (Chalkfinger) were Yitz Herstein (Israel Bond) and Irving Kaplansky (character forgotten) -- and, I think, Zygmund and Liulevicius. One memorable line was Israel Bond's: furious at a magical ring, supposedly conveying some mysterious powers, for its uselessness to his purpose at the time, Bond dashes it to the ground, huffing "Bah! This ring is nil!" The skit took place towards the end of the academic year '64-'65, all of which I spent at Chicago. Others there at that time, surely present at -- perhaps even intimately involved in -- the performance include Lance Small, Kathy Edwards, Dan Fife, and my officemate Marty Moskowitz, all of whom may well recall further details. I once had a copy of the full script, but have no idea just where to look to unearth it. Should it turn up, of course, I'll try to make it available. -- Fred [E.J. Linton] From rrosebru@mta.ca Fri Dec 9 09:34:45 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 09 Dec 2005 09:34:45 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EkiFk-0003tK-1C for categories-list@mta.ca; Fri, 09 Dec 2005 09:25:36 -0400 X-ME-UUID: 20051207223442445.6CD8A2400088@mwinf3102.me.freeserve.com Message-ID: <004f01c5fb7d$ba746080$6764893e@brown1> From: "Ronald Brown" To: References: Subject: categories: Re: Terminology question wrt fibrations of categories. Date: Wed, 7 Dec 2005 18:21:52 -0000 MIME-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 6.00.2800.1106 X-MimeOLE: Produced By Microsoft MimeOLE V6.00.2800.1106 Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 21 Thanks for these comments. I was mainly investigating the acceptance of these terms by the categorical community, to decide if I should change for the new edition of my old book. I agree with Paul's comments to me personally that it is a good idea to avoid overworked terms (like `universal'). The issue is that for a morphism f: G \to H of groupoids, the notion of quotient introduced by Philip Higgins, namely if f is full and Ob(f) is surjective, is fine. The other important notion is that f comes from an identification of objects, which in Paul's terminology would be supine (w.r.t. the opfibration Ob: Gpds \to Sets). More vivid would be H is a 0-identification of G, that is the groupoid H is then obtained from G by an identification of objects. It would tie in with other situations to say that H is induced from G by Ob(f). There is a good case for not introducing new words. Ronnie ----- Original Message ----- From: "Prof. Peter Johnstone" To: "Ronald Brown" Cc: Sent: Wednesday, December 07, 2005 9:05 AM Subject: Re: categories: Terminology question wrt fibrations of categories. > On Tue, 6 Dec 2005, Ronald Brown wrote: > > > on the other hand Paul Taylor, following Peter Johnstone, I understand, uses > > \phi is prone, supine, instead of cartesian, cocartesian > > > `Prone' and `supine' were invented by Paul Taylor; I copied them from him, > not the other way round. I'm sorry Ronnie doesn't like them; they seem to > me a very neat way of finding two words that both mean `lying > horizontally' but have an opposite handedness about them. > > Peter Johnstone > > > From rrosebru@mta.ca Fri Dec 9 09:34:45 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 09 Dec 2005 09:34:45 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EkiJd-0004Po-8B for categories-list@mta.ca; Fri, 09 Dec 2005 09:29:37 -0400 Date: Thu, 8 Dec 2005 09:36:18 -0500 (EST) From: Robert Seely To: Categories List Subject: categories: Organization vs foundations: Kreisel, Lawvere and category theory Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 22 On 29 Nov 2005, the Montreal Category seminar hosted a talk by Jean-Pierre Marquis on the interaction between Kreisel's and Lawvere's views on category theory as a foundations for mathematics. It has been suggested that many on this list who were not able to attend the talk might find its contents of interest, and so would be interested to know that the slides for the talk are available on the triples seminar webpage, at http://www.math.mcgill.ca/rags/seminar/ (scroll down to the talk itself - the direct link is http://www.math.mcgill.ca/rags/seminar/Marquis_KreiselLawvere.pdf if you prefer). The slides are fairly complete, and give a good idea of the content of the talk itself. Here is an abstract of the talk: Jean-Pierre Marquis Organization vs foundations: Kreisel, Lawvere and category theory Abstract: it is well-known that in the nineteen-sixties, Bill Lawvere proposed that category theory could serve as a foundations for mathematics and logic. Only one logician reacted officially: Georg Kreisel. In a series of notes, appendices and reviews, Kreisel developed arguments against categorical foundations. In this talk, I will take a close look at his arguments, examine whether they are still convincing and propose that Kreisel's position is still underlying most of the arguments against categorical foundations heard to this day. -- From rrosebru@mta.ca Fri Dec 9 09:34:45 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 09 Dec 2005 09:34:45 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EkiIY-0004DF-4B for categories-list@mta.ca; Fri, 09 Dec 2005 09:28:30 -0400 Message-ID: <439813A7.9000909@math.unice.fr> Date: Thu, 08 Dec 2005 12:06:15 +0100 From: "Clemens.BERGER" Reply-To: cberger@math.unice.fr Organization: Laboratoire J.-A. =?ISO-8859-1?Q?Dieudonn=E9?= X-Accept-Language: en-us, en MIME-Version: 1.0 To: categories@mta.ca Subject: categories: Re: name for a concept Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 23 Let me suggest still another terminology: For this, call a class S of maps in an arbitrary category *(co)stable* iff S is closed under composition and under (co)base change. Then call a commutative square *S-exact * (resp. *S-coexact*) iff the induced map to the pullback (resp. from the pushout) belongs to S. It is then easy to check that S-(co)exact squares compose for any (co)stable class S (which I believe is the minimal condition to impose on any reasonable distinguished class of commutative squares). In an abelian category, the class M of monos (resp. the class E of epis) is not only stable (resp. costable) but also costable (resp. stable). With this terminology, Hilton's exact squares can either be identified with the E-exact squares or with the M-coexact squares, which explains why it is a self-dual concept, cf. the first message of Michael Barr and the last message of Marco Grandis. In homotopy theory, there is the important concept of a *homotopy pullback* which is the ``homotopy invariant'' substitute for an ordinary pullback. For those who are familiar with Quillen model categories, it is very useful in practice that if a Quillen model category is *right proper* (i.e. its class of fibrations is stable), then a commutative square with two parallel fibrations is a homotopy pullback *if and only if* the square is exact with respect to the class of trivial fibrations (those fibrations which are also weak equivalences). There is of course a dual statement for homotopy pushouts in a left proper Quillen model category. With best regards, Clemens Berger. From rrosebru@mta.ca Fri Dec 9 09:34:45 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 09 Dec 2005 09:34:45 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EkiGT-0003xE-6s for categories-list@mta.ca; Fri, 09 Dec 2005 09:26:21 -0400 X-ME-UUID: 20051207224010706.AC6D45800089@mwinf3112.me.freeserve.com Message-ID: <006801c5fb7e$7dabf680$6764893e@brown1> From: "Ronald Brown" To: Subject: categories: misprint of url Date: Wed, 7 Dec 2005 22:35:13 -0000 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: 24 Claudio Hermida pointed out I had made an error, and the url should be http://www.bangor.ac.uk/~mas010/oxford281105.pdf Ronnie From rrosebru@mta.ca Fri Dec 9 09:34:45 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Fri, 09 Dec 2005 09:34:45 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EkiEh-0003og-At for categories-list@mta.ca; Fri, 09 Dec 2005 09:24:31 -0400 Subject: categories: Re: Name for a concept From: Eduardo Dubuc To: categories@mta.ca (Categories) Date: Wed, 7 Dec 2005 16:15:43 -0300 (ART) In-Reply-To: from "jean benabou" at Dec 06, 2005 11:12:54 AM X-Mailer: ELM [version 2.5 PL2] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-Id: Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 25 Jean Benabou wrote: > > (2)- In most cases the canonical map being epic is not what one really > wants. Of course Joyal assumes the category where the maps live to be a > pre-topos, then it's enough, otherwise one cannot "compose" such > squares. Do we have to rename the squares where the canonical map is a > universal epi, or those where its a universal regular epi? > very good point i suggest, since we can live with epics, strict(=regular) epis, universal such, etc etc, we should have: quasi-pullback strict(=regular) quasi pullback universal quasi-pullback of course, the useful concept being: "strict universal quasi-pullback" e.d. From rrosebru@mta.ca Wed Dec 14 16:12:51 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 14 Dec 2005 16:12:51 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Emcpt-0005lq-VF for categories-list@mta.ca; Wed, 14 Dec 2005 16:02:49 -0400 Date: Tue, 13 Dec 2005 13:33:23 +0100 (CET) From: claudio pisani Subject: categories: Preprint: Bipolar spaces To: categories@mta.ca 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: 26 Dear categorists, some of you might be interested in the following paper, now available on arXiv (http://arxiv.org/abs/math.CT/0512194) "Bipolar spaces" Claudio Pisani Abstract: Some basic features of the simultaneous inclusion of discrete fibrations and discrete opfibrations on a category A in the category of categories over A are studied; in particular, the reflections and the coreflections of the latter in the former are considered, along with a negation-complement operator which, applied to a discrete fibration, gives a functor with values in discrete opfibrations (and vice versa) and which turns out to be classical, in that the strong contraposition law holds. Such an analysis is developed in an appropriate conceptual frame that encompasses similar "bipolar" situations and in which a key role is played by "cofigures", that is components of products; e.g. the classicity of the negation-complement operator corresponds to the fact that discrete opfibrations (or in general "closed parts") are properly analyzed by cofigures with shape in discrete fibrations ("open parts"), that is, that the latter are "coadequate" for the former, and vice versa. In this context, a very natural definition of "atom" is proposed and it is shown that, in the above situation, the category of atoms reflections is the Cauchy completion of A. ----------- Of course, any comment or criticism is welcome. In particular, I would like to know which facts, apart from the formalism, sound familiar (e.g., concerning the coreflection in discrete fibrations). Best regards Claudio ___________________________________ Yahoo! Mail: gratis 1GB per i messaggi e allegati da 10MB http://mail.yahoo.it From rrosebru@mta.ca Sun Dec 18 12:30:27 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 18 Dec 2005 12:30:27 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Eo1Ee-00044r-Hh for categories-list@mta.ca; Sun, 18 Dec 2005 12:18:08 -0400 Message-Id: <200512170119.jBH1JlN17902@math-cl-n03.ucr.edu> Subject: categories: categories, Frobenius algebras, and string theory To: categories@mta.ca (categories) Date: Fri, 16 Dec 2005 17:19:47 -0800 (PST) From: "John Baez" X-Mailer: ELM [version 2.5 PL6] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 27 Dear Categorists - Here's the portion of "week224" that deals with category theory. Happy Holidays to everyone! Best, jb ....................................................................... Also available as http://math.ucr.edu/home/baez/week224.html December 14, 2005 This Week's Finds in Mathematical Physics - Week 224 John Baez This week I want to mention a couple of papers lying on the interface of physics, topology, and higher-dimensional algebra. But first, some astronomy pictures... and a bit about the mathematical physicist Hamilton! [...] Now for some mathematical physics that touches on higher-dimensional algebra. If you still don't get why topological field theory and n-categories are so cool, read this thesis: 13) Bruce H. Bartlett, Categorical aspects of topological quantum field theories, M.Sc. Thesis, Utrecht University, 2005. Available as math.QA/0512103. It's a great explanation of the big picture! I can't wait to see what Bartlett does for his Ph.D.. If you're a bit deeper into this stuff, you'll enjoy this: 14) Aaron D. Lauda and Hendryk Pfeiffer, Open-closed strings: two-dimensional extended TQFTs and Frobenius algebras, available as math.AT/0510664. This paper gives a purely algebraic description of the topology of open and closed strings, making precise and proving some famous guesses due to Moore and Segal, which can be seen here: 15) Greg Moore, Lectures on branes, K-theory and RR charges, Clay Math Institute Lecture Notes (2002), available at http://www.physics.rutgers.edu/~gmoore/clay1/clay1.html Lauda and Pfeiffer's paper makes heavy use of Frobenius algebras, developing more deeply some of the themes I mentioned in "week174". In a related piece of work, Lauda has figured out how to *categorify* the concept of a Frobenius algebra, and has applied this to 3d topology: 16) Aaron Lauda, Frobenius algebras and ambidextrous adjunctions, available as math.CT/0502550. Aaron Lauda, Frobenius algebras and planar open string topological field theories, available as math.QA/0508349. The basic idea behind all this work is a "periodic table" of categorified Frobenius algebras, which are related to topology in different dimensions. For example, in "week174" I explained how Frobenius algebras formalize the idea of paint drips on a sheet of rubber. As you move your gaze down a sheet of rubber covered with drips of paint, you'll notice that drips can merge: \ \ / / \ \ / / \ \ / / \ \ / / \ \_/ / \ / | | | | | | | | | | but also split: | | | | | | | | | | / _ \ / / \ \ / / \ \ / / \ \ / / \ \ / / \ \ In addition, drips can start: _ | | | | | | | | | | | | | | | | | | but also end: | | | | | | | | | | | | | | | | |_| In a Frobenius algebra, these four pictures correspond to four operations called "multiplication" (merging), "comultiplication" (splitting), the "unit" (starting) and the "counit" (ending). Moreover, these operations satisfy precisely the relations that you can prove by warping the piece of rubber and seeing how the pictures change. For example, there's the associative law: \ \ / / / / \ \ \ \ / / \ \ / / / / \ \ \ \ / / \ \/ / / / \ \ \ \/ / \ / / / \ \ \ / \ \ / / \ \ / / \ \_/ / \ \_/ / \ / \ / | | | | | | | | | | = | | | | | | | | | | | | | | | | | | | | | | The idea here is that if you draw the picture on the left-hand side on a sheet of rubber, you can warp the rubber until it looks like the right-hand side! There's also the "coassociative law", which is just an upside-down version of the above picture. But the most interesting laws are the "I = N" equation: \ \ / / | | | | \ \ / / | | | | \ \_/ / | | | | \ / | \ | | | | | \ | | | | | |\ \ | | | | | | \ \ | | | | | | \ \ | | | | = | | \ \ | | | | | | \ \ | | | | | | \ \| | | | | | \ | / _ \ | | \ | / / \ \ | | | | / / \ \ | | | | / / \ \ | | | | and its mirror-image version. So, the concept of Frobenius algebra captures the topology of regions in the plane! Aaron Lauda makes this fact into a precise theorem in his paper on planar open string field theories, and then generalizes it to consider "categorified" Frobenius algebras where the above equations are replaced by isomorphisms, which describe the *process* of warping the sheet of rubber until the left side looks like the right. You should look at his paper even if you don't understand the math, because it's full of cool pictures. Lauda and Pfeiffer's paper goes still further, by considering these paint stripes as "open strings", not living in the plane anymore, but zipping around in some spacetime of high dimension, where they might as well be abstract 2-manifolds with corners. Following Moore and Segal, they also bring "closed strings" into the game, which form a Frobenius algebra of their own, where the multiplication looks like an upside-down pair of pants: O O \ \ / / \ \ / / \ / | | | | | | | | | | O These topological closed strings are the subject of Joachim Kock's book mentioned in "week202"; they correspond to *commutative* Frobenius algebras. The fun new stuff comes from letting the open strings and closed strings interact. You can read more about Lauda and Pfeiffer's work at Urs Schreiber's blog: 17) Urs Schreiber, Lauda and Pfeiffer on open-closed topological strings, http://golem.ph.utexas.edu/string/archives/000680.html In fact, I recommend Schreiber's blog quite generally to anyone interested in higher categories and/or the math of string theory! ----------------------------------------------------------------------- Addendum: Here's what Urs Schreiber had to say about Frobenius algebras, modular tensor categories and string theory: John Baez wrote: [...] Following Moore and Segal, they also bring "closed strings" into the game, which form a Frobenius algebra of their own, where the multiplication looks like an upside-down pair of pants: [...] I would like to make the following general comment on the meaning of Frobenius algebras in 2-dimensional quantum field theory. Interestingly, _non_-commutative Frobenius algebras play a role even for closed strings, and even if the worldhseet theory is not purely topological. The archetypical example for this is the class of 2D TFTs invented by Fukuma, Hosono and Kawai. There one has a non-commutative Frobenius algebra which describes not the splitting/joining of the entire worldsheet, but rather the splitting/joining of edges in any one of its dual triangulations. It is the _center_ of (the Morita class of) the noncommutative Frobenius algebra decorating dual triangulations which is the commutative Frobenius algebra describing the closed 2D TFT. One might wonder if it has any value to remember a non-commutative Frobenius algebra when only its center matters (in the closed case). The point is that the details of the non-commutative Frobenius algebra acting in the "interior" of the world sheet affects the nature of "bulk field insertions" that one can consider and hence affects the (available notions of) n-point correlators of the theory, for n > 0. This aspect, however, is pronounced only when one switches from 2D topological field theories to conformal ones. The fascinating thing is that even 2D "conformal" field theories are governed by Frobenius algebras. The difference lies in different categorical internalization. The Frobenius algebras relevant for CFT don't live in Vect, but in some other (modular) tensor category, usually that of representations of some chiral vertex operator algebra. It is that ambient tensor category which "knows" if the Frobenius algebra describes a topological or a conformal field theory (in 2D) - and which one. Of course what I am referring to here is the work by Fjelstad, Froehlich, Fuchs, Runkel, Schweigert and others. I can recommend their most recent review which will appear in the Streetfest proceedings. It is available as math.CT/0512076. The main result is, roughly, that given any modular tensor category with certain properties, and given any (symmetric and special) Frobenius algebra object internal to that category, one can construct functions on surfaces that satisfy all the properties that one would demand of an n-point function of a 2D (conformal) field theory. If we define a field theory to be something not given by an ill-defined path integral, but something given by its set of correlation functions, then this amounts to constructing a (conformal) field theory. This result is achieved by first defining a somewhat involved procedure for generating certain classes of functions on marked surfaces, and then proving that the functions generated by this procedure do indeed satisfy all the required properties. In broad terms, the prescription is to choose a dual triangulation of the marked worldsheet whose correlation function is to be computed, to decorate its edges with symmetric special Frobenius algebra objects in some modular tensor category, to decorate its vertices by product and coproduct morphisms of this algebra, to embed the whole thing in a certain 3-manifold in a certain way and for every boundary or bulk field insertion to add one or two threads labeled by simple objects of the tensor category which connect edges of the chosen triangulation with the boundary of that 3-manifold. Then you are to hit the resulting extended 3-manifold with the functor of a 3D TFT and hence obtain a vector in a certain vector space. This vector, finally, is claimed to encode the correlation function. This procedure is deeply rooted in well-known relations between 3-(!)-dimensional topological field theory, modular functors and modular tensor categories and may seem very natural to people who have thought long enough about it. It is already indicated in Witten's paper on the Jone's polynomial, that 3D TFT (Chern-Simons field theory in that case) computes conformal blocks of conformal field theories on the boundaries of these 3-manifolds. To others, like me in the beginning, it may seem like a miracle that an involved and superficially ad hoc procedure like this has anything to do with correlations functions of conformal field theory in the end. In trying to understand the deeper "meaning" of it all I played around with the idea that this prescription is really, to some extent at least, the "dual" incarnation of the application of a certain 2-functor to the worldsheet. Namely a good part of the rough structure appearing here automatically drops out when a 2-functor applied to some 2-category of surfaces is "locally trivialized". I claim that any local trivialization of a 2-functor on some sort of 2-category of surface elements gives rise to a dual triangulation of the surface whose edges are labeled by (possibly a generalization of) a Frobenius algebra object and whose vertices are labeled by (possibly a generalization of) product and coproduct operations. There is more data in a locally trivialized 2-functor, and it seems to correctly reproduce the main structure of bulk field insertions as appearing above. But of course there is a limit to what a _2_-functor can know about a structure that is inherently 3-dimensional. I have begun outlining some of the details that I have in mind here: http://golem.ph.utexas.edu/string/archives/000697.html This has grown out of a description of gerbes with connective structure in terms of transport 2-functors. Note that in what is called a _bundle_ gerbe we also do have a certain product operation playing a decisive role. Bundle gerbes can be understood as "pre-trivializations" of 2-functors to Vect: http://golem.ph.utexas.edu/string/archives/000686.html and the product appearing is one of the Frobenius products mentioned above. For a bundle gerbe the coproduct is simply the inverse of the product, since this happens to be an isomorphism. The claim is that 2-functors to Vect more generally give rise to non-trivial Frobenius algebras when locally trivialized. This is work in progress and will need to be refined. I thought I'd mention it here as a comment to John's general statements about how Frobenius algebras know about 2-dimensional physics. I am grateful for all kinds of comments. Here's the paper Urs refers to: 21) Ingo Runkel, Jens Fjelstad, Jurgen Fuchs and Christoph Schweigert, Topological and conformal field theory as Frobenius algebras, available as math.CT/0512076. ----------------------------------------------------------------------- Quote of the Week: Here's how you do it: First you're obtuse, Then you intuit, Then you deduce! - Garrison Keillor ----------------------------------------------------------------------- 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 http://math.ucr.edu/home/baez/this.week.html From rrosebru@mta.ca Sun Dec 18 12:30:27 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 18 Dec 2005 12:30:27 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Eo1FU-00046e-Sl for categories-list@mta.ca; Sun, 18 Dec 2005 12:19:00 -0400 Date: Sun, 18 Dec 2005 00:09:31 +0100 From: Joachim Kock Subject: categories: Re: nerves In-reply-to: <43906B25.9020802@math.upenn.edu> X-Sender: kock@127.0.0.1:55110 To: categories@mta.ca Message-id: MIME-version: 1.0 Content-type: text/plain; charset=iso-8859-1 Content-transfer-encoding: QUOTED-PRINTABLE References: <43906B25.9020802@math.upenn.edu> Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 28 > Does anyone know a reference for > grothedieck's characterization for when a simplicial set > is the nerve of a groupoid > or if anyone observed it earlier? >=20 > jim The earliest written testemony I know of is in Seminaire Bourbaki, 19= 61: A. Grothendieck, Technique de descente et theoremes d'existence en= =20 geometrie algebrique, III: Preschemas quotients. Seminaire Bourbak= i=20 t.12, 1960/61, exp. 212. In Section 4, categories are characterised as presheaves on Delta tha= t take amalgamated sums over [0] to fibre products. (Delta =3D skeleto= n of finite non-empty ordinals). Groupoids are characterised as presheave= s on Phi taking amalgamated sums to fibre products, where Phi is the symme= tric version of Delta, i.e. skeleton for non-empty phinite sets and any ma= ps. (Delta and Phi are not Grothendieck's notation.) Also the terminology 'nerve of a category' is usually attributed to Grothendieck, but it is actually not used in the above Expose'. (Of course, 'nerve of a covering' goes much further back -- I think t= o=20 Cech in the 1930s.) The details probably should have been in SGA1 expose' VII (which was = never written), and appeared instead in Section 2 of J. Giraud, Methode de = la descente, Bull. Soc. Math. France Mem., 1964. (Giraud was supposed t= o write Expose' VII, but the manuscript got longer and longer, less and= =20 less geometric, and was delayed for these reasons, and finally he dec= ided=20 to publish it separately instead. (He says something like this in th= e=20 introduction to the long memoir.)) Cheers, Joachim. ---------------------------------------------------------------- Joachim Kock Departament de Matem=E0tiques -- Universitat Aut=F2noma de Barcelona Edifici C -- 08193 Bellaterra (Barcelona) -- ESPANYA Phone: +34 93 581 32 50 Fax: +34 93 581 27 90 http://mat.uab.es/~kock/ ---------------------------------------------------------------- From rrosebru@mta.ca Sun Dec 18 16:15:25 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sun, 18 Dec 2005 16:15:25 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Eo4sP-0003U5-78 for categories-list@mta.ca; Sun, 18 Dec 2005 16:11:25 -0400 Date: Mon, 19 Dec 2005 04:31:24 +1100 (EST) From: Michael Johnson Message-Id: <200512181731.jBIHVOou005413@vesuvius.ics.mq.edu.au> To: categories@mta.ca Subject: categories: AMAST CFP (February deadline) Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 29 The 2006 edition of AMAST (Algebraic Methodology and Software Technology) has been scheduled to follow CT'06 so that travelling category theorists with an interest in computer science have time to proceed from Nova Scotia to Estonia. AMAST in the past has had some excelent category theoretic papers, and if you're interested in submitting, the call for papers is below. Best (and festive!) wishes, Mike. -------------------------- CALL FOR PAPERS 11th International Conference on Algebraic Methodology and Software Technology, AMAST '06 colocated with MPC '06 Kuressaare, Estonia, 5-8 July 2006 http://cs.ioc.ee/mpc-amast06/amast/ Background The major goal of the AMAST conferences is to promote research that may lead to the setting of software technology on a firm, mathematical basis. This goal is advanced by a large international cooperation with contributions from both academia and industry. The virtues of a software technology developed on a mathematical basis have been envisioned as being capable of providing software that is (a) correct, and the correctness can be proved mathematically, (b) safe, so that it can be used in the implementation of critical systems, (c) portable, i.e., independent of computing platforms and language generations, and (d) evolutionary, i.e., it is self-adaptable and evolves with the problem domain. The previous conferences were held in Iowa City, Iowa, USA (1989, 1991 and 2000), Twente, The Netherlands (1993), Montreal, Canada (1995), Munich, Germany (1996), Sydney, Australia (1997), Manaus, Amazonia, Brazil (1998), Reunion Island, France (2002) and Stirling, UK (2004, colocated with MPC' 04). The 2006 conference will be held at Kuressaare, Estonia, colocated with MPC '06. The conference series has become widely known for disseminating academic and industrial achievements within the broad AMAST areas of interest. Through these meetings AMAST has attracted an international following among researchers and practitioners interested in software technology, programming methodology and their algebraic and logical foundations. Important dates * Submission of abstracts: 27 January 2006 * Submission of full papers: 3 February 2006 * Notification of authors: 17 March 2006 * Camera-ready version: 14 April 2006 Topics Topics of interest include, but are not limited to, the following: SOFTWARE TECHNOLOGY: * systems software technology * application software technology * concurrent and reactive systems * formal methods in industrial software development * formal techniques for software requirements, design * evolutionary software/adaptive systems PROGRAMMING METHODOLOGY: * logic programming, functional programming, object paradigms * constraint programming and concurrency * program verification and transformation * programming calculi * specification languages and tools * formal specification and development case studies ALGEBRAIC AND LOGICAL FOUNDATIONS: * logic, category theory, relation algebra, computational algebra * algebraic foundations for languages and systems, coinduction * theorem proving and logical frameworks for reasoning * logics of programs * algebra and coalgebra SYSTEMS AND TOOLS (for system demonstrations or ordinary papers): * software development environments * support for correct software development * system support for reuse * tools for prototyping * component based software development tools * validation and verification * computer algebra systems * theorem proving systems Submission Two kinds of submissions are solicited for this conference: technical papers and system demonstrations. Papers may report academic or industrial progress, and papers which deal with both are especially well-regarded. Submission is in two stages. Abstracts (plain text) must be submitted by 27 January 2006. Full papers (pdf) adhering to the llncs style and not longer than 15 pages (6 pages for system demonstrations) must be submitted by 3 February 2006. The web-based submission system will open in early January 2006. 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. All papers will be refereed by the programme committee, and will be judged based on their significance, technical merit, and relevance to the conference. Publication As in the past the proceedings of AMAST '06 will be published in the Lecture Notes in Computer Science series of Springer-Verlag. In the past the best papers from AMAST have been published in a special issue of the journal Theoretical Computer Science and it is expected that this practice will be continued for AMAST '06. Programme Committee Chairs Michael Johnson, Macquarie University (co-chair) Varmo Vene, University of Tartu (co-chair) AMAST steering committee Egidio Astesiano, Universita degli Studi di Genova Robert Berwick, MIT Zohar Manna, Stanford University Michael Mislove, Tulane University Anton Nijholt, University of Twente Maurice Nivat, Universite Paris 7 Charles Rattray, University of Stirling Teodor Rus, University of Iowa Giuseppe Scollo, Universita degli Studi di Verona Michael Sintzoff, Universite Catholique de Louvain Jeannette Wing, Carnegie Mellon University Martin Wirsing, Ludwig-Maximilians-Universitaet Muenchen Michael Johnson, Macquarie University (chair) Venue Kuressaare (pop. 16000) is the main town on Saaremaa, the second-largest island of the Baltic Sea. Kuressaare is a charming seaside resort on the shores of the Gulf of Riga highly popular with Estonians as well as visitors to Estonia. The scientific sessions of MPC/AMAST 2006 will take place at Saaremaa Spa Hotel Meri, one among the several new spa hotels in the town. The social events will involve a number of sites, including the 14th-century episcopal castle. Accommodation will be at Saaremaa Spa Hotels Meri and Ruutli. To get to Kuressaare and away, one must pass through Tallinn (pop. 402000), Estonia's capital city. Tallinn is famous for its picturesque medieval Old Town, inscribed on UNESCO's World Heritage List. Local organizers MPC/AMAST 2006 is organized by Institute of Cybernetics, a research institute of Tallinn Univ. of Technology. The local organizers are Tarmo Uustalu (chair), Monika Perkmann, Juhan Ernits, Ando Saabas, Olha Shkaravska, Kristi Uustalu. Contact email addresses: mike(at)ics.mq.edu.au, varmo(at)cs.ut.ee. From rrosebru@mta.ca Mon Dec 19 09:04:08 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 19 Dec 2005 09:04:08 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EoKby-0006wL-6L for categories-list@mta.ca; Mon, 19 Dec 2005 08:59:30 -0400 Message-Id: <6.2.1.2.1.20051219124506.0397cbb0@mat.uc.pt> X-Mailer: QUALCOMM Windows Eudora Version 6.2.1.2 Date: Mon, 19 Dec 2005 12:46:10 +0000 To: categories@mta.ca From: Maria Manuel Clementino Subject: categories: Postdoctoral Research Positions Mime-Version: 1.0 Content-Type: text/plain; charset="us-ascii"; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 30 Postdoctoral Research Positions for 2006/2007 The Centre for Mathematics of the University of Coimbra (CMUC) accepts applications for one-year postdoctoral positions in all areas of Mathematics. CMUC has been recently classified as a research unit of excellence by an international funding panel. In particular, the Centre welcomes applications in the following areas: Computational Mathematics & Scientific Computing. Integer Programming & Combinatorial Optimization. Group and Algebras Representations. Category Theory. Multilinear Algebra, Matrix Theory & Combinatorics. Nonparametric Statistics. Nonlinear Time-Series Modelling. Partial Differential Equations. Free Boundary Problems and Nonlinear Applied Problems. Quantum Groups. Symplectic and Poisson Geometry. The positions obey to the portuguese scholarship system (http://www.fct.mces.pt). The salary will be determined by the candidate's qualifications and experience and can vary between 17940 and 24720 EUR a year (tax free). The positions include benefits and a professional travel allowance. There are no teaching duties associated to the positions, which should start by September/October 2006. Applicants must have (or soon have) a Ph. D. in Mathematics, preferably obtained after January 1, 2003, and a good command of English. Applications will be accepted until January 31, 2006. Applicants should send a curriculum vitae (publication list included), a statement of research interests (one page maximum), and at least two letters of recommendation by regular mail or e-mail to: Centre for Mathematics - University of Coimbra Apartado 3008, 3001-454 Coimbra, Portugal cmuc@mat.uc.pt http://www.mat.uc.pt/~cmuc From rrosebru@mta.ca Tue Dec 20 21:34:25 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 20 Dec 2005 21:34:25 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EoshE-0000VY-Dc for categories-list@mta.ca; Tue, 20 Dec 2005 21:23:12 -0400 X-ME-UUID: 20051210033409471.0B7B91C00082@mwinf1207.wanadoo.fr Date: Sat, 10 Dec 2005 04:51:20 +0100 Subject: categories: Terminology Content-Type: text/plain; charset=US-ASCII; format=flowed Mime-Version: 1.0 (Apple Message framework v543) To: Categories From: jean benabou Content-Transfer-Encoding: 7bit Message-Id: <398E964C-6930-11DA-B8BE-000393B90F2C@wanadoo.fr> Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 31 I have seen in this mail that the suggestion of Taylor and Johnstone to replace cartesian and cocartesian maps by prone and supine ones begins to be accepted. When I first saw that suggestion, I was so amazed that I thought it was a joke, and not such a good one. I still hope it is no more than that. But, just in case, and before it is too late, I want to say that I am very strongly opposed to such changes for many reasons: linguistic, mathematical, and ethical, which I am ready to explain in detail if I am asked to do so. From rrosebru@mta.ca Tue Dec 20 21:34:24 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Tue, 20 Dec 2005 21:34:24 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Eoskt-0000e3-96 for categories-list@mta.ca; Tue, 20 Dec 2005 21:26:59 -0400 Date: Mon, 19 Dec 2005 19:29:55 -0400 (AST) From: Bob Rosebrugh To: categories Subject: categories: list postings intermittent Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: RO X-Status: X-Keywords: X-UID: 32 The categories moderator will be out of regular email contact from December 21 to 30, 2005. Postings submitted to Categories during that period will be distributed intermittently. Best wishes, Bob Rosebrugh From rrosebru@mta.ca Wed Dec 21 10:02:48 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Wed, 21 Dec 2005 10:02:48 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Ep4UV-0000mR-47 for categories-list@mta.ca; Wed, 21 Dec 2005 09:58:51 -0400 Date: Tue, 20 Dec 2005 20:16:24 -0600 From: Peter May Message-Id: <200512210216.jBL2GOcC016863@math.uchicago.edu> To: categories@mta.ca Subject: categories: Right on, Jean! Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 33 It's funny, but exactly that question of terminology (prone, supine, etc) came up a few weeks ago in some joint work with a student here. I made the same case to him, almost exactly, that Jean Benabou just made. And one other objection: I'd like category theory no longer to be regarded as nonsense in this country --- it still is in many quarters, as I could easily prove --- and such terminology is not exactly helpful to the cause! Peter May ps: Then again, I don't much like using the overused words cartesian and cocartesian. From rrosebru@mta.ca Mon Dec 26 11:29:11 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 26 Dec 2005 11:29:11 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EquBX-0003B3-Gm for categories-list@mta.ca; Mon, 26 Dec 2005 11:22:51 -0400 To: categories@mta.ca Subject: categories: 2006 Programme on Logic and Algorithms - Workshops Message-Id: <20051222201406.238544A9C5@cs.rice.edu> Date: Thu, 22 Dec 2005 14:14:06 -0600 (CST) From: myv@cs.rice.edu (MYV) Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 34 During the first half of 2006, the Isaac Newton Institute for Mathematical Sciences at Cambridge, UK, will hold a Special Programme on Logic and Algorithms. The programme will include six workshops: 9 - 13 January Finite and Algorithmic Model Theory (A Satellite Meeting at Durham) 27 February - 3 March Logic and Databases 20 - 24 March Mathematics of Constraint Satisfaction: Algebra, Logic and Graph Theory (A Satellite Meeting at Oxford) 10 - 13 April New Directions in Proof Complexity 8 - 12 May Constraints and Verification 3 - 7 July Games and Verification The workshops are open to participation. See http://www.newton.cam.ac.uk/programmes/LAA/ws.html To join the mailing list, see http://www.newton.cam.ac.uk/programmes/LAA/list.html For further information, contact vardi at cs dot rice dot edu. Moshe Vardi From rrosebru@mta.ca Mon Dec 26 11:29:11 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 26 Dec 2005 11:29:11 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Equ7I-00034H-0R for categories-list@mta.ca; Mon, 26 Dec 2005 11:18:28 -0400 Mime-Version: 1.0 (Apple Message framework v746.2) To: categories@mta.ca Message-Id: <783D14D6-29BF-43FF-945F-D2F789AAC5FA@mcs.vuw.ac.nz> Content-Type: multipart/alternative; boundary=Apple-Mail-7-123886765 From: Rob Goldblatt Subject: categories: tenured position available Date: Thu, 22 Dec 2005 09:36:32 +1300 Content-Transfer-Encoding: 7bit Content-Type: text/plain;charset=US-ASCII;delsp=yes;format=flowed Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 35 Please circulate in your department, forward to any interested parties, and copy to any relevant lists. Apologies for multiple postings. ============================================= LECTURER IN MATHEMATICS SCHOOL OF MATHEMATICS, STATISTICS AND COMPUTER SCIENCE VICTORIA UNIVERSITY OF WELLINGTON, NEW ZEALAND Applications are invited for the tenured position of Lecturer in Mathematics which will become available in the School of Mathematics, Statistics and Computer Science in July 2006. (The position of Lecturer corresponds approximately to that of Assistant Professor in North America). Candidates with research interests in any area of mathematics are welcome to apply. The School's Mathematics Group is strongly committed to excellence in research and in teaching mathematics at all levels. It runs a wide range of undergraduate and graduate courses, and contains some of New Zealand's leading research mathematicians in several areas of pure and applied mathematics. More detailed information is on the School's website at http:// www.mcs.vuw.ac.nz/info/vacancies/sa0569m . Academic enquiries about the position may be directed to Professor Rod Downey (Rod.Downey@mcs.vuw.ac.nz) or Professor Rob Goldblatt (Rob.Goldblatt@mcs.vuw.ac.nz). Applications close 17 February 2006. Application information is available from the HR Adviser, Faculties of Science and Architecture & Design, email: science-appoint@vuw.ac.nz ; telephone +64 4 463 5100. Reference SA0569M. From rrosebru@mta.ca Mon Dec 26 11:29:11 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 26 Dec 2005 11:29:11 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Equ8K-00036O-Gp for categories-list@mta.ca; Mon, 26 Dec 2005 11:19:32 -0400 From: SOS 2006 Organisers To: categories@mta.ca Date: Thu, 22 Dec 2005 14:57:43 +1100 Message-Id: <1051222035743.30542@cse.unsw.edu.au> Subject: categories: SOS 2006 - Call for Papers Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 36 First Call for Papers: Structural Operational Semantics 2006 A Satellite Workshop of CONCUR 2006 August 26, 2006, Bonn, Germany http://www.cse.unsw.edu.au/~rvg/SOS2006 Aim: Structural operational semantics (SOS) provides a framework for giving operational semantics to programming and specification languages. A growing number of programming languages from commercial and academic spheres have been given usable semantic descriptions by means of structural operational semantics. Because of its intuitive appeal and flexibility, structural operational semantics has found considerable application in the study of the semantics of concurrent processes. Moreover, it is becoming a viable alternative to denotational semantics in the static analysis of programs, and in proving compiler correctness. Recently, structural operational semantics has been successfully applied as a formal tool to establish results that hold for classes of process description languages. This has allowed for the generalisation of well-known results in the field of process algebra, and for the development of a meta-theory for process calculi based on the realization that many of the results in this field only depend upon general semantic properties of language constructs. This workshop aims at being a forum for researchers, students and practitioners interested in new developments, and directions for future investigation, in the field of structural operational semantics. One of the specific goals of the workshop is to establish synergies between the concurrency and programming language communities working on the theory and practice of SOS. Moreover, it aims at widening the knowledge of SOS among postgraduate students and young researchers worldwide. Specific topics of interest include (but are not limited to): * programming languages * process algebras * higher-order formalisms * rule formats for operational specifications * meaning of operational specifications * comparisons between denotational, axiomatic and SOS * compositionality of modal logics with respect to operational specifications * congruence with respect to behavioural equivalences * conservative extensions * derivation of proof rules from operational specifications * software tools that automate, or are based on, SOS. Papers reporting on applications of SOS to software engineering and other areas of computer science are welcome. History: The first SOS Workshop took place in August 2004 in London as one of the satellite workshops of CONCUR 2004, and was attended by over 30 participants [http://www.cs.aau.dk/~luca/SOS-WORKSHOP/]. The second SOS Workshop occurred in July 2005 in Lisbon as a satellite workshop of ICALP 2005, and attracted 19 submissions [http://www.cs.le.ac.uk/events/SOS2005/]. INVITED SPEAKERS: * Robin Milner (Cambridge, UK; joint invited speaker with EXPRESS 2006) * Bartek Klin (Sussex, UK) PAPER SUBMISSION: We solicit unpublished papers reporting on original research on the general theme of SOS. Prospective authors should register their intention to submit a paper by uploading a title and abstract via the workshop web page by: *** Friday 26 May 2006. *** Papers should take the form of a dvi, postscript or pdf file in ENTCS format [http://www.entcs.org/], whose length should not exceed 15 pages (not including an optional "Appendix for referees" containing proofs that will not be included in the final paper). We will also consider 5-page papers describing tools to be demonstrated at the workshop. Submissions from PC members are allowed. Proceedings: Preliminary proceedings will be available at the meeting. The final proceedings of the workshop will appear as a volume in the ENTCS series. If the quality and quantity of the submissions warrant it, the co-chairs plan to arrange a special issue of an archival journal devoted to full versions of selected papers from the workshop. IMPORTANT DATES: * Submission of abstract: Friday 26 May 2006 * Submission: Sunday 4 June 2006, midnight GMT * Notification: Wednesday 28 June 2006 * Final version: Friday 14 July 2006 * Workshop: Saturday 26 August 2006 * Final ENTCS version: Friday 29 September 2006. PROGRAMME COMMITTEE Rocco De Nicola (Florence, IT) Wan Fokkink (Amsterdam, NL) Rob van Glabbeek (NICTA, AU, co-chair) Reiko Heckel (Leicester, UK) Matthew Hennessy (Sussex, UK) Ugo Montanari (Pisa, IT) Peter Mosses (Swansea, UK, co-chair) MohammadReza Mousavi (Eindhoven, NL) David Sands (Chalmers, SE) Irek Ulidowski (Leicester, UK) Shoji Yuen (Nagoya, JP) CONTACT: sos2006@cs.stanford.edu WORKSHOP ORGANISERS: Rob van Glabbeek National ICT Australia Locked Bag 6016 University of New South Wales Sydney, NSW 1466 Australia Peter D. Mosses Department of Computer Science Swansea University Singleton Park Swansea SA2 8PP United Kingdom From rrosebru@mta.ca Mon Dec 26 11:29:11 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 26 Dec 2005 11:29:11 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Equ65-00031s-3m for categories-list@mta.ca; Mon, 26 Dec 2005 11:17:13 -0400 Subject: categories: Re: Terminology From: Eduardo Dubuc To: categories@mta.ca (Categories) Date: Wed, 21 Dec 2005 17:04:11 -0300 (ART) In-Reply-To: <398E964C-6930-11DA-B8BE-000393B90F2C@wanadoo.fr> from "jean benabou" at Dec 10, 2005 04:51:20 AM X-Mailer: ELM [version 2.5 PL2] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-Id: Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 37 I very strongly agree with J. Benabou's comments about "prone" and "supine", and P. May's opinion that "I'd like category theory no longer to be regarded as nonsense in this country --- it still is in many quarters, as I could easily prove --- and such terminology is not exactly helpful to the cause!" I recall P. Johnstone that he himself named his book "Elephant Book" because every body has different version of what a topos is, reflecting only one of the many aspects of the concept. Names like "Prone" and "Supine" correspond (with luck) to only one of the many aspects of the concept of cartesian and its dual (in a sense) cocartesian. Also, there is a clear ethical aspect involved when a stablished terminology that has been historically introduced by particular people suffers a move to be eliminated and reeplaced by another. But, coming back to the question above, i am also against the habit to name a new mathematical concept with words that have a precise meaning in everyday language (as prone, supine, etc). Presisely, I do not know what does it mean exactly "Cartesian" (has something to do with Descartes ...), but I know presisely what it is a "Cartesian arrow" (in mathematics). Colorful terminology taken from everyday language is an strong indication to serious mathematicians that the subject should no be taken seriously (see for example the claims of "Catastrofe Theory" as opposed to the sober "Classification of singularities of C-\infty mappings", and a lot of similar examples). As P May points out, " . . . such terminology is not exactly helpful to the cause!". The meaning of a mathematical concept should be given by the concept itself, and not by the connotation that its name has in everyday language. From rrosebru@mta.ca Mon Dec 26 11:29:11 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 26 Dec 2005 11:29:11 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Equ4p-0002zi-7H for categories-list@mta.ca; Mon, 26 Dec 2005 11:15:55 -0400 Message-ID: <014801c60659$872bf9e0$e77d893e@brown1> From: "Ronald Brown" To: References: <200512210216.jBL2GOcC016863@math.uchicago.edu> Subject: categories: Re: Right on, Jean! Date: Wed, 21 Dec 2005 18:08:20 -0000 MIME-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 38 For the revision of my old topology book, I want to keep the analogies which the fibrations of categories concept brings out very well - I am coming (very late!) to the view that this is an important part of `categories for the working mathematician' (the concept, not necessarily the book). In the general topology part of the book, I have final topologies with respect to a function, and also identification maps, as in Bourbaki. So (unless anyone can quickly come with anything better) I have decided on replacing (a) `universal morphism of groupoids f: G \to H' as in the current text and Philip Higgins' book, and which uses an overused word `universal', by (a') ` f gives H the final structure with respect to G and Ob(f)' (b) `under these circumstances, f is a 0-identification morphism if also Ob(f) is surjective'. I initially (!) wanted to say `f is a 0-final morphism' instead of (a') but Tim pointed out it was initial in an appropriate category! Another possibility is `H has the induced structure w.r.t Ob(f)', and to use the res/ind terminology from representation theory and Mackey functors. Comments on these issues welcome. But the aim is to use terminology which has associations and emphasises analogies. The new title will be `Topology and groupoids', which seems better to reflect the content. It will (all being well) be available as a print and ebook, with the ebook in color and hyper-reference. Ronnie www.bangor.ac.uk/r.brown ----- Original Message ----- From: "Peter May" To: Sent: Wednesday, December 21, 2005 2:16 AM Subject: categories: Right on, Jean! > > It's funny, but exactly that question of terminology > (prone, supine, etc) came up a few weeks ago in some > joint work with a student here. I made the same case > to him, almost exactly, that Jean Benabou just made. > And one other objection: I'd like category theory no > longer to be regarded as nonsense in this country --- > it still is in many quarters, as I could easily prove --- > and such terminology is not exactly helpful to the cause! > > Peter May > > ps: Then again, I don't much like using the overused > words cartesian and cocartesian. > > > From rrosebru@mta.ca Mon Dec 26 11:29:11 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 26 Dec 2005 11:29:11 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EquA4-00039B-RJ for categories-list@mta.ca; Mon, 26 Dec 2005 11:21:20 -0400 X-ME-UUID: 20051222081357410.641D25800084@mwinf3108.me.freeserve.com Message-ID: <001901c606ce$ee6d2dc0$c8cb4c51@brown1> From: "Ronald Brown" To: Subject: categories: Terminology re fibrations and opfibrations of categories Date: Thu, 22 Dec 2005 08:07:43 -0000 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: 39 To add to my previous email, I'd like reactions to the following terminology: Let P: X \to B be a functor. A morphism u: x \to y in X is cofinal w.r.t. P, and y is the P-final object w,r,t u and P , if ... (and here we have the usual notion of cocartesian). Dually, u is coinitial, and x is the initial object w.r.t u and P if ... (and here we have the usual notion of cartesian). In situations where P is understood, we can then talk about cofinal and coinitial morphisms, and structures or objects or (in my case, groupoids). An advantage is that the direction of the notion and its dual should be clear. If f=P(u), I would then write \bar{f}: x \to f_*(x) in the first case, and \underline{f}: f^*(y) \to y in the second. I would also call f_*(x) the object induced by f. What is a handy name for f^*(y)? The restriction of y by f? All these notions occur for modules, crossed modules, ...... and relate to change of base. Ronnie www.bangor.ac.uk/r.brown From rrosebru@mta.ca Mon Dec 26 11:29:11 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 26 Dec 2005 11:29:11 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EquCR-0003Cn-5J for categories-list@mta.ca; Mon, 26 Dec 2005 11:23:47 -0400 Date: Fri, 23 Dec 2005 10:46:30 +0100 (CET) From: Jiri Adamek To: categories net Subject: categories: idempotent completion Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 40 I would be grateful for getting the earliest reference to the fact that for two small categories T and S the corresponding functor-categories into Set are equivalent iff T and S have the same idempotent (= Cauchy) completion. One can find this in a russian paper: "Morita equivalent categories" by S. V. Polin, Vestnik Mosk. Univ., 1974, no.2, 41-45 xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx alternative e-mail address (in case reply key does not work): J.Adamek@tu-bs.de xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx From rrosebru@mta.ca Thu Dec 29 13:12:26 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 29 Dec 2005 13:12:26 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Es1Dc-0002rB-Mt for categories-list@mta.ca; Thu, 29 Dec 2005 13:05:36 -0400 Reply-To: marta.bunge@mcgill.ca In-Reply-To: From: "Marta Bunge" To: categories@mta.ca Subject: categories: RE: idempotent completion Date: Mon, 26 Dec 2005 15:57:47 -0500 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 41 Dear Jirka, The result appears in my 1966 thesis ("Categories of Set-valued functors", U. of Pennsylvania), and in print (with an arbitrary closed category base V and categories relative to V) in the following 1969 paper, translated into Russian in 1972. Marta Bunge, Relative Functor Categories and Categories of Algebras. J.of Algebra 11 (1969) 64-101. Russian translation in : Mathematics: Periodical collections of Translations of Foreign Articles, Vol.16, Izdat. "Mir", Moscow(1972) 11-46, MR 50, #12532. Cordially, Marta ************************************************ Marta Bunge Professor Emerita Dept of Mathematics and Statistics McGill University 805 Sherbrooke St. West Montreal, QC, Canada H3A 2K6 Office: (514) 398-3810 Home: (514) 935-3618 marta.bunge@mcgill.ca http://www.math.mcgill.ca/~bunge/ ************************************************ >From: Jiri Adamek >To: categories net >Subject: categories: idempotent completion >Date: Fri, 23 Dec 2005 10:46:30 +0100 (CET) > >I would be grateful for getting the earliest reference to the fact >that for two small categories T and S the corresponding >functor-categories into Set are equivalent iff T and S have the same >idempotent (= Cauchy) completion. One can find this in a russian paper: > >"Morita equivalent categories" by S. V. Polin, Vestnik Mosk. Univ., >1974, no.2, 41-45 > > >xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx >alternative e-mail address (in case reply key does not work): >J.Adamek@tu-bs.de >xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx > > From rrosebru@mta.ca Thu Dec 29 13:12:26 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 29 Dec 2005 13:12:26 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Es1F8-0002uZ-9U for categories-list@mta.ca; Thu, 29 Dec 2005 13:07:10 -0400 Date: Mon, 26 Dec 2005 15:56:42 -0500 (EST) From: Peter Freyd Message-Id: <200512262056.jBQKugLo026545@saul.cis.upenn.edu> To: categories@mta.ca Subject: categories: Re: idempotent completion Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 42 Jiri asks: I would be grateful for getting the earliest reference to the fact that for two small categories T and S the corresponding functor- categories into Set are equivalent iff T and S have the same idempotent (= Cauchy) completion. The fact that a category and its idempotent completion have equivalent functor categories was certainly known very early. It does not appear in the first book on category theory (1964) but the lemma that proves it, to wit, that idempotent-complete cats form a full reflective subcategory of the relevant category (COSCANECOF) appears on page 61 (which is 18 pages before any mention of reflective subcats and 48 pages before the first mention of functor categories -- see www.tac.mta.ca/tac/reprints/articles/3/). That book was devoted to the additive setting. On page 119 one finds the additive notion, "amenable", corresponding to the condition of idempotents splitting. The full subcat of small projectives in the functor category in the additive setting is dual to the amenable closure of the domain category -- thus providing an instant proof that if two cats have equivalent additive functor categories then their amenable closures are equivalent. The non-additive case is easier: the full subcat of indecomposable projectives in the category of set- valued functos is dual to the idempotent completion of the domain category. From rrosebru@mta.ca Thu Dec 29 13:12:26 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 29 Dec 2005 13:12:26 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Es1GM-0002wf-7z for categories-list@mta.ca; Thu, 29 Dec 2005 13:08:26 -0400 Subject: categories: Re: Terminology re fibrations and opfibrations of categories From: Eduardo Dubuc To: categories@mta.ca Date: Mon, 26 Dec 2005 18:57:48 -0300 (ART) In-Reply-To: <001901c606ce$ee6d2dc0$c8cb4c51@brown1> from "Ronald Brown" at Dec 22, 2005 08:07:43 AM X-Mailer: ELM [version 2.5 PL2] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-Id: Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 43 Concerning Ronnie wanderings about terminology around the word FINAL, the following is pertinent: I am just writing a paper with Luis Espannol where we need to develop (the basic part of the theory of cartesian and cocartesian arrows) for families we use the following terminology: consider a functor U: C ---> S, then: 1) a family in C Z _i ---> X over R_i ---> S is FINAL iff: given S ---> T = UY such that there exists Z_i --->Y over R_i ---> S ---> T (that is, R_i ---> S ---> T lifts), then there exists a unique X ---> Y over S ---> T (that is, S ---> T lifts). For topological spaces this is the usual Bourbaki notion of final topology. When U is not understood, we call this "U-FINAL" Notice that for single arrows, we have (proved in the SGA on fibered categories) Z ---> X is final iff it is cocartesian and cocartesian arrows compose 2) a family in C Z _i ---> X over R_i ---> S is SURJECTIVE iff: the family R_i ---> S is an strict (or regular) epimorphic family in S Our aim is to prove under some natural and minimal assumptions: Z _i ---> X is strict epimorphic iff it is final surjective All this is already done Here the leading examples are the topological spaces and the quasitopological spaces in the sense of Spanier (and the whole theory of concrete quasitopoi over S = Sets) From rrosebru@mta.ca Thu Dec 29 13:12:26 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Thu, 29 Dec 2005 13:12:26 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1Es1CZ-0002pB-Jy for categories-list@mta.ca; Thu, 29 Dec 2005 13:04:31 -0400 Message-ID: <43B048D4.8020601@cs.stanford.edu> Date: Mon, 26 Dec 2005 11:47:32 -0800 From: Vaughan Pratt Reply-To: pratt@cs.stanford.edu MIME-Version: 1.0 To: Categories Subject: categories: Re: terminology References: In-Reply-To: Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 44 Without taking sides on the prone/supine terminology question, I do have a strong reaction to the Benabou/May/Dubuc concern that respect for a field is undermined by its adoption of frivolous terminology. This may be a valid concern for a young field like category theory, but for a more mature subject such as physics, a more relevant concern is the undermining of the ability to poke fun at oneself by the fear of not being taken seriously. Has the adoption of frivolous nomenclature for quarks ("strange," "charm," "beauty" and even "quark" itself) diminished in any way the world's respect for quarks and their investigators? And what of computational topology? Should we turn a blind eye to whether Scott is sober, and substitute a more genteel euphemism for his bottom? Vaughan Pratt From rrosebru@mta.ca Sat Dec 31 10:27:33 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 31 Dec 2005 10:27:33 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EshXX-0007gE-UO for categories-list@mta.ca; Sat, 31 Dec 2005 10:17:00 -0400 From: Nikita Danilov MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Message-ID: <17332.13426.996827.974494@gargle.gargle.HOWL> Date: Thu, 29 Dec 2005 22:09:38 +0300 To: Categories Subject: categories: Re: terminology Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 45 Vaughan Pratt writes: > Without taking sides on the prone/supine terminology question, I do have > a strong reaction to the Benabou/May/Dubuc concern that respect for a > field is undermined by its adoption of frivolous terminology. > > This may be a valid concern for a young field like category theory, but > for a more mature subject such as physics, a more relevant concern is > the undermining of the ability to poke fun at oneself by the fear of not > being taken seriously. > > Has the adoption of frivolous nomenclature for quarks ("strange," > "charm," "beauty" and even "quark" itself) diminished in any way the > world's respect for quarks and their investigators? There indeed are drawbacks whenever scientific terms are contrary to the centuries old tradition not taken from Greek or Latin languages (that, thanks to their very regular and flexible system of word formation are so suitable for taxonomies) shared by many cultures. For one thing, words of existing languages are not in one to one mapping, and then a term from contemporary language may be not culturally neutral (consider silly naming wars for transuranium elements). On the other hand, I stopped using "co-product" after more than one person with the background in classical languages read it as "copro-duct". > > And what of computational topology? Should we turn a blind eye to > whether Scott is sober, and substitute a more genteel euphemism for his > bottom? > > Vaughan Pratt Nikita. From rrosebru@mta.ca Sat Dec 31 10:27:33 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 31 Dec 2005 10:27:33 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EshdD-00002D-KZ for categories-list@mta.ca; Sat, 31 Dec 2005 10:22:51 -0400 Date: Fri, 30 Dec 2005 17:29:15 +0100 Mime-Version: 1.0 (Apple Message framework v543) Content-Type: text/plain; charset=ISO-8859-1; format=flowed Subject: categories: Terminology again From: jean benabou To: Categories Content-Transfer-Encoding: quoted-printable Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: RO X-Status: X-Keywords: X-UID: 46 In a previous mail I said that I was strongly opposed to the=20 replacement of "cartesian" and "co-cartesian" maps by "prone" and=20 "supine" ones for linguistic, mathematical and ethical reasons which I=20= was ready to explain in detail if I was asked to do so. I have waited a few days to see what reactions I would get. So far, my=20= position has been supported by Peter May and Eduardo Dubuc on ethical=20 and/or linguistic grounds, and Keith Harbaugh has asked me what=20 "ethical" issue is involved, and more generally to clarify my position=20= on the mailing list. Although the ethical issues are for me the most important, I shall=20 postpone them to another mail, if the moderator of this list permits me=20= to do so, and I shall concentrate to-day on the mathematical an=20 linguistic reasons of my opposition. I am no linguist, but I am a mathematician, so THE MATHEMATICS WILL=20 COME FIRST, and I would like to make this text more "palatable" and=20 less "negative" by introducing some genuinely new ideas, which some of=20= you might find of interest, and which are relevant in this debate. =A71- DEFINITIONS Let P: C ---> B be a a functor. 1.1- A map v of C is VERTICAL (forP) if P(v) is an identity. I=20 denote by V(P) or simply V the subcategory of C which has the same=20 objects as C and as maps the vertical ones. Every identity is vertical, and if in a commutative triangle in C two=20 of the maps are vertical so is the third. 1.2- Let f be a map f of C. We shall say that f is : (i) CARTESIAN if for every pair (g,b) with g in C , b in B, and=20 P(g)=3DP(f).b there exists a unique map h in C such that: P(h)=3Db = and =20 g=3Df.h. (ii) PRECARTESIAN If the previous condition is satisfied only when b=20 is an identity, (iii) HORIZONTAL if every vertical map is orthogonal to f . (that=20 would probably be "prone" in the proposed new terminology) I shall denote by K(P), PreK(P) and H(P) the classes of maps of C=20 which are cartesian, precartesian and horizontal, and abbreviate by K,=20= PreK, and H if P is fixed. 1.3 - The functor P is a fibration (resp. a PREFIBRATION) if for every=20= pair (b,X) where b is a map of B, and X an object of C and=20 P(X)=3DCodom(b) there exists a cartesian (resp.precartesian) map f: Y=20= --->X such that P(f)=3Db 1.4 - If X is an object of C and b: J--->P(X) we denote Pl(b,X) the=20 category of P-LIFTINGS of b with codomain X with objects the maps f:=20 Y--->X of C such that P(f)=3Db, a morphism from f to f': Y'--->X is a=20= vertical map v: Y--->Y' such that f.v=3Df' . I shall say that P is a HOMOTOPY PREFIBRATION if all the categories =20 Pl(b,X) are connected (which of course imply non empty), the motivation=20= of the name is that any two liftings are "homotopic". 1.5 - Let P: C ---> B and P': C' ---> B be two functors and F; C=20= ---> C' be a functor over B i.e. such that P'.F=3DP. Such an F = obviously=20 preserves and reflects vertical maps for P and P'. Moreover if f: Y=20 --->X is in Pl(b,X) , F(f): F(Y) ---> F(X) is in P'l(b,F(X)) hence F=20 induces functors Fl(b,X): Pl(b,X) --->P'l(b,F(X)) , f l--->F(f) , for all=20 "compatible" pairs (b,X) I shall say that F is a CARTESIAN FUNCTOR if all the functors Fl(b,X)=20 are final. (No assumption is made on P or P'). =A72-REMARKS "EN VRAC" ABOUT THESE DEFINITIONS (or, let's do a little = bit=20 of mathematics!) 2.1- Let P: C --->B be a functor (we assume NOTHING on P) Then: (i) Every cartesian map is horizontal i.e. K(P) is contained in H(P). The converse need not be true, for example if all vertical maps are=20 iso's i.e. all the fibers of P are groupo=EFds, then all maps of C are=20= horizontal, and "co-horizontal" And one can construct a P where C and B=20= are finite posets where no map, except of course the identities, is=20 pre-cartesian or pre-cocartesian, let alone cartesian or cocartesian (ii) THEOREM. If P is a homotopy- prefibration, then cartesian=20 coincides with horizontal, i.e. H(P)=3DK(p). (It is not completely trivial) The definitions 1.4 and 1.5 are genuinely new, and might seem=20 surprising, the following remarks will give a very small idea of what=20 can be done with them 2.2- A cartesian functor preserves precartesian maps : Because f: Y --->X is in PreK(P) iff it is a final object of =20 Pl(P(f),X) , and final functors preserve final objects However if P and P' are are arbitrary functors such a preservation is=20 not enough to insure that F is cartesian because there might not be=20 "enough" precartesian maps in C. But cartesian functors have so far=20 NEVER been used except between prefibrations, and in that case our=20 definition coincides with the usual one because we have: 2.3- If P and P' are prefibrations, F is cartesian iff it preserves=20= precartesian maps.(It suffices in fact that P is a prefibration) I like to make the following "analogy" : if S and T are topological=20 spaces a continuous function f: S --->T preserves convergent=20 sequences, if X is metrisable, this is enough to insure the continuity=20= of f 2.4 - Cartesian functors are closed under composition, and every=20 equivalence over B is cartesian. In fact we have much better, namely. 2.5 - If a functor F over B has a left adjoint then F is cartesian 2.6 - Homotopy-prefibrations are stable by composition. This seems "harmless" and trivial, but it is neither. It is well known=20= that fibrations are stable by composition, but it is probably a little=20= less well known, because I have never seen a statement to that effect,=20= that prefibrations ARE NOT . 2.7 - Homotopy-prefibrations (h-p) are special cases of cartesian=20 functors, because P. C --->B is a h-p iiff it is a cartesian functor:=20= (C,P) --->(B,IdB). (This of course is no longer true if h-p is replaced by prefibration or=20= fibration) =46rom this it follows that if F:(C,P) --->(C',P') is cartesian and = P'=20 is a h-p so is P. 2.8 - An important feature of h-p is that pointwise Kan extensions=20 along such P's can be computed fiberwise. Moreover this property=20 characterizes h-p' s. In particular such a P is final iff all its fibers are connected, and=20 it is flat iff it's fibers are cofiltered. The previous results are special cases of properties true for arbitrary=20= cartesian functors. 2.9 - REMARK : Homotopy prefibrations are but ONE example of =20 MEANINGFUL generalizations of fibrations. I have considered many=20 others, all with important mathematical examples, here are some: (for a=20= functor P: C --->B) (i) The categories Pl(b,X) are filtered (ii) Each connected component of such a category has a final object (iii) Each connected component is filtered In (ii) and (ii) P is not even a homotopy prefibration, but in all=20 these cases the general definition of cartesian functor given in 1.5 =20 is the "correct" one and gives the expected results. =A73 LINGUISTICO-MATHEMATICAL REMARKS 3.1 - OK, let us try "prone" for "cartesian", what about the=20 precartesian maps, "preprone" ? They have nothing to do with a=20 weakening of orthogonality to V(P), which we shall examine in 3.3. What=20= about cartesian functors, "prone functors"? What about maps which are=20 both cartesian and cocartesian, such that e.g. the iso's, prone and=20 supine? A very uncomfortable position you'll grant me. I am no acrobat,=20= I tried it, I hurt my back and stomach, had to stand up, and ended=20 up...vertical! 3.2 - The proposed terminology is based ON A BIG MATHEMATICAL MISTAKE,=20= namely: confusing cartesian and horizontal, which in general do NOT=20 coincide, as shown in 2.1. Unless of course there no other functors but=20= fibrations, or if there are, the terminology should not be compatible=20 with them. Well I, and probably other persons, think that there are,=20 know that there are, and that they deserve to be studied, were it only=20= to have a better understanding of fibrations. In 2.9 I gave a few=20 examples of such functors. If there were ONLY fibrations, how would=20 one express the fact that a prefibration where all the fibers are=20 groupoids is a fibration? 3.3 - Even for fibrations there are interesting maps which are neither=20= vertical nor cartesian and that one might want to study. Let me give an=20= example. Both cartesianness and horizontality assume the existence and=20= uniqueness of maps satisfying certain conditions. What about those=20 where we drop existence and keep uniqueness. Following Peter Freyd's suggestion, let me call them=20 quasi-cartesian(QK), and quasi-horizontal(QH),and see what they are. A=20= map f: Y --->X is QK (rep. QH) iff for every parallel pair =20 (g,g'): Z=3D=3D=3D>Y coequalized by f, if P(g)=3DP(g') (resp.if = g.v=3Dg'.v for=20 some vertical map v) then g=3Dg' Even in the case of fibrations, where K=3DH, QK is only contained in QH=20= but not equal.This can be seen in the most trivial case, where B=3D1, = and=20 all maps are vertical. A map f is QK iff it is a mono, it is QH iff =20 for every pair of maps (g,g') WHICH CAN BE EQUALIZED, fg=3Dfg' implies = =20 g=3Dg'. Now if cartesian=3Dprone, QK will have to be "quasi-prone", a strange=20 position again, but never mind. However, how should we call QH ? 3.4- I can speak, read, and write a little bit of English, but I am=20 French and might someday have the preposterous idea to lecture on=20 fibered categories in French. Of course only in France, and to an=20 audience uniquely composed of french persons. Perhaps MM Taylor and=20 Johnstone, could suggest adequate french translations for prone and=20 supine, which I can't seem to find. And they should be ready to do the=20= same thing for German, Italian, Spanish, and many other languages. No such problems with cartesian of course, because cartesian.... is=20 cartesian is cartesian is cartesian! 3.5- By now many thousands of pages have been written in various=20 languages using "cartesian", and many hundreds are being prepared, or=20 ready to be published, using the same word. What should be done with=20 all that past or future rubbish, now that we have received THE LIGHT=20 and the WORD(S)? =A74 TEMPORARY CONCLUSION I apologize for such a long mail, but I wanted also to show, among=20 other things , that it is possible to handle new and relevant=20 mathematical notions by introducing a SINGLE new word, namely;=20 "homotopy prefibration" , which has a clear intuitive content, and=20 moreover is easy to translate in most languages. I have given many arguments to explain my position, and I have many=20 more. But for the moment, I'd like to know the arguments of the persons=20= in favor of these changes, PRINCIPALLY, of course, those of Paul Taylor=20= and Peter Johnstone. If it is only the "joke" aspect, I want to add=20 that I do also like jokes, very much, perhaps not the same as theirs..=20= I even used to compete with Sammy, who was an expert, about who'd know=20= some jokes the other didn't. When this mail was almost completely finished, I found the reaction of=20= Vaughan Pratt from which I quote: "Has the adoption of frivolous nomenclature for quarks ("strange," "charm," "beauty" and even "quark" itself) diminished in any way the world's respect for quarks and their investigators?" I want to be clear on that matter. I have no objection to "frivolous"=20 naming of NEW concepts by the person or persons who DISCOVERED or=20 INVENTED them. But I object VERY STRONGLY to "renaming" well=20 established concepts, used for more than 40 years by the mathematical=20 community, even if the new names were NOT frivolous, and especially if=20= such a renaming is made by persons who have made no MAJOR contribution to the development of the field of FIBERED CATEGORIES. As a side remark, I have no problem whatsoever to translate in French :=20= "strange", "charm", "beauty", "quark", "sober" or "bottom". And to be=20 "frivolous", even if it's not so easy in a foreign language,=20 "homotopy's bottom" came ages before Scott's, and "Galois connection"=20 ages before the "french" one. Since my english is not too good, in particular I knew only "the other"=20= meaning of "supine", I'll borrow, a bit freely, from "a good author" I=20= admire a lot, and remind that: Men gave names to many animals In the beginning, in the beginning Men gave names to many animals In the beginning, long time ago. Best wishes to all, Jean From rrosebru@mta.ca Sat Dec 31 10:27:33 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 31 Dec 2005 10:27:33 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EshYU-0007hu-EX for categories-list@mta.ca; Sat, 31 Dec 2005 10:17:58 -0400 Subject: categories: Re: terminology From: Eduardo Dubuc To: categories@mta.ca (Categories) Date: Thu, 29 Dec 2005 20:17:36 -0300 (ART) In-Reply-To: <43B048D4.8020601@cs.stanford.edu> from "Vaughan Pratt" at Dec 26, 2005 11:47:32 AM X-Mailer: ELM [version 2.5 PL2] 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: 47 We should not put everything in the same bag !! "strange," "charm," "beauty" and even "quark" itself are beautiful and poetic names to refer to objects or concepts which precisely we do not want to associate any precise meaning in everyday language, and on the other hand, the objects or concepts are introduced whith those names. "prone/supine" are all the contrary, they intent to reflect in everyday language just one aspect of an existing concept which has many, and more important, they are used in place of a well stablished name. all this has nothing to do with "young field" as opposed to "mature subject" silly names (if any) in physics would be as bad as in any other subject do not confuse things, I found the "Scott is sober" an exelent example of humor that does not undermine respect for the field. Another exelent example that comes to my mind is M. Barr's "The point of the empty set" edubuc > > Without taking sides on the prone/supine terminology question, I do have > a strong reaction to the Benabou/May/Dubuc concern that respect for a > field is undermined by its adoption of frivolous terminology. > > This may be a valid concern for a young field like category theory, but > for a more mature subject such as physics, a more relevant concern is > the undermining of the ability to poke fun at oneself by the fear of not > being taken seriously. > > Has the adoption of frivolous nomenclature for quarks ("strange," > "charm," "beauty" and even "quark" itself) diminished in any way the > world's respect for quarks and their investigators? > > And what of computational topology? Should we turn a blind eye to > whether Scott is sober, and substitute a more genteel euphemism for his > bottom? > > Vaughan Pratt > > From rrosebru@mta.ca Sat Dec 31 10:27:33 2005 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Sat, 31 Dec 2005 10:27:33 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EshZN-0007kk-I7 for categories-list@mta.ca; Sat, 31 Dec 2005 10:18:53 -0400 From: vs27@mcs.le.ac.uk To: Categories Subject: categories: Re: terminology Date: 30 Dec 2005 01:16:46 +0000 X-Mailer: Prayer v1.0.12 X-Originating-IP: [125.0.160.177] Content-Type: text/plain; format=flowed; charset=ISO-8859-1 Sender: cat-dist@mta.ca Precedence: bulk Message-Id: Status: O X-Status: X-Keywords: X-UID: 48 On Dec 29 2005, Vaughan Pratt wrote: > Without taking sides on the prone/supine terminology question, I do have > a strong reaction to the Benabou/May/Dubuc concern that respect for a > field is undermined by its adoption of frivolous terminology. > Dear Vaughan, as everybody has a say. Just my views. I prefer some nomenclature that sounds mathematical, rather than based on the name of a friend or a private joke. (may be i don't understand all the jokes ?) Also in any case one should avoid renaming existing concepts, that is just not fair. Good opportunity to wish happy new year to everybody. From rrosebru@mta.ca Mon Jan 2 10:00:26 2006 -0400 Return-path: Envelope-to: categories-list@mta.ca Delivery-date: Mon, 02 Jan 2006 10:00:26 -0400 Received: from Majordom by mailserv.mta.ca with local (Exim 4.52) id 1EtQ4N-0007PI-9n for categories-list@mta.ca; Mon, 02 Jan 2006 09:49:51 -0400 In-Reply-To: References: Mime-Version: 1.0 (Apple Message framework v623) Content-Type: text/plain; charset=US-ASCII; format=flowed Message-Id: Content-Transfer-Encoding: 7bit From: David Yetter Subject: categories: Re: Terminology again + Note from moderator Date: Sat, 31 Dec 2005 22:01:54 -0600 To: Categories Sender: cat-dist@mta.ca Precedence: bulk Status: O X-Status: X-Keywords: X-UID: 49 [Note from moderator: This is to let you know that I am invoking the (not recently used) 48 hour rule for this subject: postings received by noon on Wednesday will be sent; after that the discussion may obviously continue, but not on the list. Best wishes to all for 2006, Bob Rosebrugh] On 30 Dec 2005, at 10:29, jean benabou wrote: > I want to be clear on that matter. I have no objection to "frivolous" > naming of NEW concepts by the person or persons who DISCOVERED or > INVENTED them. But I object VERY STRONGLY to "renaming" well > established concepts, used for more than 40 years by the mathematical > community, even if the new names were NOT frivolous, and especially if > such a renaming is made by persons who have made no MAJOR > contribution to the development of the field of FIBERED CATEGORIES. > I will second Jean's remarks excerpted above, with a sole exception: I have no objection to the renaming of a well-established concept in honor of the person(s) who discovered or invented them (or, if a second name is attached, who first made its importance clear). Best wishes to all for the new year, David Yetter