From rrosebru@mta.ca Mon May 1 15:18:50 2006 -0300 Date: Mon, 1 May 2006 13:56:24 +0930 From: David Roberts To: categories@mta.ca Subject: Re: gr-stacks in the literature MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 8bit Status: O X-Status: X-Keywords: X-UID: 1 Thanks for replies. I am familiar with Breen's work, and the correspondence between crossed modules and gr-stacks. I ws thinking more along the lines of "a smooth gr- chart K (or whatever it is called) for a differentiable gr-stack G is a representable surjection K --> G satisfying the properties..." (with some 2- cartesian squares following). Such a thing might be in Champs Algebriques , in the category of schemes, but I haven't got it at the library. -- David Roberts Pure Mathematics University of Adelaide South Australia, 5005 You know we all became mathematicians for the same reason: we were lazy. -Max Rosenlicht(1949) From rrosebru@mta.ca Mon May 1 15:20:09 2006 -0300 Date: Mon, 1 May 2006 09:53:47 +0100 (BST) From: "Prof. Peter Johnstone" To: categories Subject: Re: Does duality categorify? MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: X-Keywords: X-UID: 57 On Sat, 29 Apr 2006, Vaughan Pratt wrote: > As an initial guess: "categories", with dualizer Set, which is both a > category and a distributive category. So is CAT(C,Set) a distributive > category? And if so, is DCAT(CAT(C,Set),Set) equivalent to C? And what > about the enriched case V-DCAT(V-CAT(C,V),V)? > There's a result related to this (though not, I think, actually containing it) in my paper with Andre Joyal on continuous categories and exponentiable toposes (JPAA 25, 1982). We showed that there is an equivalence (which is really a duality, but the arrows on one side have been reversed) between quasi-injective toposes (that is, toposes which occur as retracts of presheaf toposes) and the categories that occur as their categories of points, which are continuous categories satisfying a certain size restriction. Peter Johnstone From rrosebru@mta.ca Mon May 1 15:21:05 2006 -0300 Date: Mon, 1 May 2006 12:46:54 +0100 From: K C H Mackenzie To: categories@mta.ca Subject: Re: dualities MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 8bit Status: O X-Status: X-Keywords: X-UID: 58 Quoting Peter Freyd : > On the subject of favorite dualities: > > Surely the most important are the self-dualities and the most > important of these (so important we stop noticing it as we age) is the > category of finite-dimensional vector spaces over a given field. Something on this has been done. Duality for vector bundle objects in the category of Lie groupoids was done by Jean Pradines in 1988, and is part of the fundamental work on symplectic groupoids. The cotangent bundle $T^*G$ of any Lie groupoid $G$ has a groupoid structure with base the dual of $AG$, the Lie algebroid of $G$, and Pradines' construction realizes this as the dual of the tangent prolongation $TG$ of $G$. A double vector bundle (in the sense of Ehresmann) is a particular instance of a vector bundle in the category of Lie groupoids. Pradines' duality can be applied to such a structure in two ways, and these do not commute. If $D$ is a double vector bundle over vector bundles $A$ and $B$, each of which is a vector bundle over a manifold $M$, then $D$ can be dualized over $A$ and over $B$. These dualization operations generate the dihedral group of order 6. See Duality and triple structures', pp455--481 of The breadth of symplectic and Poisson geometry', (Weinstein Festschrift), Progr. Math., Birkh\"auser Boston, 2005. Alfonso Gracia-Saz and I are preparing a paper on the duality of $n$-fold vector bundles. Details and references for the double case can be found in my General Theory of Lie groupoids and Lie algebroids', Cambridge, 2005, Chapter 9. Whether categlorification would add anything to this I do not know. Kirill Mackenzie From rrosebru@mta.ca Mon May 1 20:56:56 2006 -0300 Date: Mon, 1 May 2006 15:06:07 -0400 (EDT) From: Michael Barr To: Categories list Subject: Re: dualities MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: X-Keywords: X-UID: 112 First, let me say I have avoided contributing to this thread because I don't understand what Vaughan is asking. He knows, as well as anyone, since he put them on the map, about Chu categories. He knows about *-autonomous categories as well. So what is the question, really? The simplest answer to Mamuka's question is the duality between vector spaces (over any field, including the 2 element field) and linearly compact vector spaces. In the case of a finite field, linear compactness is the same as the ordinary topological kind. One proof of this fact is that the category of finite dimensional spaces is self-dual and if two categories are dual, the inductive completion of one is dual to the projective completion of the other. For finite dimensional vector spaces, the inductive completion is vector spaces and the projective completion is linearly compact ones. Another example is the obvious duality between finite sets and finite boolean algebras that gives Stone duality on one hand and the duality between Set and CABA on the other, depending which one you complete which way. Most examples I am aware of of self-dualities are Chu categories (or chu categories). And if V_k is the category of k-vector spaces, then Chu(V_k,k) (an object is a pair of spaces and a bilinear pairing into k) is *-autonomous, as is chu(V_k,k) of separated extensional pairs. Peter's example is one of the very few *-autonomous categories I cannot relate to Chu. Complete (say inf) semi-lattices is another. Michael From rrosebru@mta.ca Mon May 1 20:58:06 2006 -0300 From: "Ronnie Brown" To: Subject: Re: dualities Date: Mon, 1 May 2006 21:02:33 +0100 MIME-Version: 1.0 Content-Type: text/plain;format=flowed;charset="iso-8859-1";reply-type=original Content-Transfer-Encoding: 7bit Status: O X-Status: X-Keywords: X-UID: 113 Peter Freyd writes on Pontrjagin duality. I would like to mention the generalisation given in (with P.J. HIGGINS and S.A. MORRIS), Countable products of lines and circles: their closed subgroups, quotients and duality properties'', {\em Math. Proc. Camb. Phil. Soc.} 78 (1975) 19-32. One point made is that a duality is not necessarily inherited by closed subgroups and Hausdorff quotients. If it is, it is called a strong duality. Then strong duality is inherited by closed subgroups and Hausdorff quotients! I have taught the classification of closed subgroups of R^n in an analysis course. It is a nice result, and the sums you can set use duality in a nice way - treatment borrowed from Bourbaki. Ronnie Brown > Next is Pontryagin's: the category of locally compact groups. The > original Pontryagin duality easily generalizes: the category of > locally compact modules over a given commutative ring is self-dual. > (In the non-commutative case one also obtains a duality but not a > self-duality -- unless, of course, the ring is self-dual.) A corollary > is that the category of discrete left R-modules is dual to the > category of compact right R-modules From rrosebru@mta.ca Tue May 2 21:19:33 2006 -0300 Date: Mon, 01 May 2006 22:39:35 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: Categories list Subject: Re: dualities Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: O X-Status: X-Keywords: X-UID: 114 Michael Barr wrote: > First, let me say I have avoided contributing to this thread because I > don't understand what Vaughan is asking. He knows, as well as anyone, > since he put them on the map, about Chu categories. He knows about > *-autonomous categories as well. So what is the question, really? Duality is of necessity between categories, and involves associating an object (say an algebra or space) of one category with its dual in another, or in the same category in the self-dual case. Downstairs, i.e. in 2-CAT. By "categorifying duality" I meant a duality of 2-categories in which one associates an object (this time a category rather than an algebra) of one 2-category with its dual in another, with the functors being reversed (op) as opposed to the natural transformations (co). Upstairs, i.e. in 3-CAT. Regarding Chu, I was going to respond that the Chu construction works downstairs with categories (from my usual V=Set perspective), or at most V-categories, categories enriched in V, as objects of the 2-category V-CAT. However if the enriched Chu construction can be organized to allow V to be a 2-category, with Chu(V,k) then being a 3-category, maybe Mike is on to a promising approach (though it's not clear that's what he actually meant). It's an aspect of Chu spaces I know next to nothing about however. My first guess would be that it (moving Chu up into 3-CAT) ought to work fine, with the caveat that the simple notion of Stone topology as a totally disconnected compact Hausdorff topology would turn into the proverbial thousand flowers---there's far more room for such stuff in 3-CAT than 2-CAT. (Actually there's also a lot of unexplored such territory even just in ordinary Chu(Set,3).) Along those lines, Peter Johnstone's mention of quasi-injective toposes dual to continuous categories in his 1982 paper with Joyal is surely just scratching the surface of the possible permutations and combinations up there in 3-CAT. Peter's > example is one of the very few *-autonomous categories I cannot relate to > Chu. Complete (say inf) semi-lattices is another. Oh, but complete inf semilattices are one of the most elegant self-dualities of chupology. They embed in Chu(Set,2) as those biextensional Chu spaces (biextensional = no repeated rows or columns), of any cardinality, such that the set of rows is closed under arbitrary AND (think of them as bit vectors) and likewise for the set of columns. No other conditions. Using OR instead of AND for both rows and columns gives sup semilattices. But that was in my previous post, where I also mentioned that XOR in place of OR, at least for finite Chu spaces, embeds FinVct_{GF(2)}. In either case the symmetry of the conditions makes the self-duality immediate. All that is in the 1999 Coimbra notes I cited. Vaughan From rrosebru@mta.ca Tue May 2 21:20:15 2006 -0300 Subject: FMCS'06 reminder From: Pieter Hofstra To: categories@mta.ca Content-Type: text/plain Date: Tue, 02 May 2006 10:47:19 -0600 Mime-Version: 1.0 Content-Transfer-Encoding: 7bit Status: O X-Status: X-Keywords: X-UID: 115 Dear All, This is to remind you that abstracts for the FMCS'06 meeting in Kananaskis, June 7-9, are due on Friday, May 21st. Since we have almost reached the maximum number of participants for the event, it is especially important for those who are planning to come but who haven't confirmed this yet to send me an e-mail as soon as possible. (The list of confirmed participants can be found on the website below.) Due to the limited capacity of the facilities of the Field Stations, where the meeting will be held, we unfortunately cannot guarantee a place for people who attempt a last minute registration. More information about the meeting (which will be held right after the Categories and Semigroup Workshop and the CMS 2006 Summer Meeting) can be found here: http://pages.cpsc.ucalgary.ca/~robin/FMCS/FMCS_06/CatMeetings.html If you are planning to come but are in doubt whether your participation is confirmed, please check the participants list at: http://pages.cpsc.ucalgary.ca/~robin/FMCS/FMCS_06/parti.html If you have any questions about any of the meetings, do not hesitate to send me an e-mail at hofstrap@cpsc.ucalgary.ca From rrosebru@mta.ca Tue May 2 21:21:23 2006 -0300 Subject: Re: dualities To: categories@mta.ca (categories) Date: Tue, 2 May 2006 15:05:24 -0700 (PDT) From: "John Baez" MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Status: O X-Status: X-Keywords: X-UID: 116 Hi - > First, let me say I have avoided contributing to this thread because I > don't understand what Vaughan is asking. He asked if dualities categorify. I guess he meant something like this: There are lots of interesting examples of a pair of categories C,D together with an object c in C and an object d in D such that hom(-,c): C -> D and hom(-,d): D -> C are part of an equivalence of categories. In the nicest examples, c and d are in some sense the same mathematical entity regarded as living in two different categories - a "schizophrenic object", in the words of Harold Simmons. So, can we find equally nice examples where C and D are instead 2-categories? In particular, can we find examples where C and D are 2-categorical generalizations of the 1-categorical examples we already know? In particular, he suggested taking the example where C is the category of finite distributive lattices and finding an analogous example where C is the 2-category of (maybe finite, in some sense?) distributive categories. For more on "schizophrenic objects", Peter Johnstone's review of Clark and Davies' "Natural dualities for the working algebraist" makes good reading: http://north.ecc.edu/alsani/ct99-00(8-12)/msg00116.html From rrosebru@mta.ca Wed May 3 17:38:30 2006 -0300 From: "jpradines" To: Subject: re: duality Date: Wed, 3 May 2006 11:44:03 +0200 MIME-Version: 1.0 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Status: O X-Status: X-Keywords: X-UID: 117 %%%%%%%%%%%%% Kirill wrote: ................. Something on this has been done. Duality for vector bundle objects in the category of Lie groupoids was done by Jean Pradines in 1988, and is part of the fundamental work on symplectic groupoids. The cotangent bundle $T^*G$ of any Lie groupoid $G$ has a groupoid structure with base the dual of $AG$, the Lie algebroid of $G$, and Pradines' construction realizes this as the dual of the tangent prolongation $TG$ of $G$. ............... Kirill Mackenzie %%%%%%%%%%%%%%%%%%% Since Kirill alluded to my note entitled "Remarque sur le groupoide = cotangent de Weinstein-Dazord", published in CRAS, 306, 557-560 = (1988), which I did not intend to do, I think I've better add some more = comments. The origin of this Note is an attempt to understand (and categorify!) = the discovery by Alan Weinstein, and (more or less independantly) by = various authors, that, given a smooth groupoid G (in the sense of = Ehresmann) with base B, its cotangent bundle T*G owns a canonical = groupoid structure (apart from the abelian group bundle structure of = course), which, when G is a Lie group, reduces to the action groupoid = describing the coadjoint action (the base of this groupoid is then the = dual of the Lie algebra; in the general case it is the dual vector = bundle of what I introduced under the name of Lie algebroid). The construction of these authors made an extensive use of the = symplectic structure of the cotangent bundle, and of symplectic duality = and orthogonality, describing the graph of the groupoid composition law = by means of Lagrangian submanifolds. Actually they prove more, since = they show that this groupoid is a symplectic groupoid (in a sense that = has also certainly to be better categorified, since it is not easily = described as an internal groupoid in some known category, but that's = another story, which I don't want to tackle here). I was amazed by this result, since, while it is obvious, by (covariant!) = functoriality of T, that the tangent bundle TG has a groupoid structure = with base TB (which describes the action of G on "vertical vectors", = more precisely on the Lie algebroid), one would better expect for the = cotangent bundle some kind of co-groupoid structure. Finally I realized that the result has nothing to do with the symplectic = structure, and extends in a natural way for what is called in my Note = "vector bundle groupoids", which means internal groupoids in the = category of vector bundles (without any given extra structure ). Such a = vector bundle groupoid has a canonically defined dual object in the same = category. =20 The construction is made "by hand", and I am not really satisfied with = it (though it oversimplifies the genuine one). The point is of course = that the"duality" in the category of vector bundles is easy to define = and handle only when the base of the bundles remains fixed, while the = construction has to cope with general vector bundle morphisms, with at = least three different bases: B, G and the set of composable arrows. Notably I was not able to discover some simple general duality relation = between the nerves of the groupoid and of its dual, as one would like. So I am still amazed with the result, which should become "obvious" in a = better adapted framework to be discovered. Best regards. Jean Pradines From rrosebru@mta.ca Wed May 3 17:39:22 2006 -0300 Date: Wed, 3 May 2006 15:58:50 +0200 (MEST) From: Peter Schuster To: categories@mta.ca Subject: MAP summer school. Call for participation. MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: X-Keywords: X-UID: 165 Call for participation. The MAP group (see http://www.disi.unige.it/map/) MAP = Mathematics, Algorithms, Proofs organises a one week summer school in Genova (Italy) from Monday 28th August 2006 to Saturday 2nd September 2006. The programme is as follows: Thierry Coquand (Goteborg) : Proof analysis 3h Erich Kaltofen (NCSU, USA) : Computer algebra 6h Henri Lombardi (Besancon) : Constructive commutative algebra 6h Marie-Francoise Roy (Rennes) : History of algorithmic real algebra 3h Francis Sergeraert (Grenoble) : Constructive homologica algebra 6h Helmut Schwichtenberg (Munich): Constructive analysis 6h Wednesday afternoon : free Saturday morning : included An on-line registration form is available from http://www.disi.unige.it/map/. ***** The deadline for registration is 31 May 2006. ***** Successful registration will be notified automatically by email. The registration fee of 180 Euro will include double-room shared accommodation from Sunday (arrival) to Saturday (departure) as well as lunch and dinner from Monday to Friday. The fee is to be payed in cash (Euro) upon arrival (no credit cards). See the aforementioned web page for more details. The organising committee: Henri Lombardi, Herve Perdry, Giuseppe Rosolini, Peter Schuster, and John Abbott From rrosebru@mta.ca Wed May 3 17:41:23 2006 -0300 Date: Wed, 3 May 2006 08:42:31 -0400 (EDT) From: Phil Scott To: categories@mta.ca Subject: ad for postdoc at Ottawa U. MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: X-Keywords: X-UID: 166 ------------------------------------------------------------------------- Research Fellow/Postdoc position in Logic and Computation, University of Ottawa The Logic Group in the Department of Mathematics and Statistics at the University of Ottawa is looking to hire a research fellow/postdoc beginning in September, 2006 (negotiable). This position is in any area of category theory, categorical logic, and theoretical computer science which complements the interests of our group. Research fellows / postdocs will participate in the activities of the Logic and Foundations of Computation Group. This group includes faculty and students from several different Ottawa-area universities. In the Math Department, the Logic Group currently includes 3 faculty members (R. Blute, P. Scott, P.Hofstra, who is joining the group in January, 2007), and 6 graduate students. For more information about our team, see http://www.site.uottawa.ca/~phil/lfc/ The research fellowships/postdocs are initially for one year, with a possible renewal for a second year. The annual salary is $40,000 Can. Duties include research and the teaching of two one-semester courses. Normally teaching is one course (in Fall and Winter terms) although possibly one course in summer may be available. Potential applicants should contact one of us: Philip Scott (phil@site.uottawa.ca) Richard Blute (rblute@mathstat.uottawa.ca) immediately by email to indicate their interest. They should then also send a curriculum vitae, a research plan, and arrange for three confidential letters of recommendation, with one addressing teaching, to be sent to Professor Michel Racine, Chairman, Department of Mathematics and Statistics, University of Ottawa, Ottawa, ON Canada, K1N 6N5. Applicants are also encouraged to include up to three copies of their most significant publications. The above position has just suddenly become available and interested persons should contact us as soon as possible. P. Scott, R. Blute, and P. Hofstra From rrosebru@mta.ca Wed May 3 17:43:28 2006 -0300 Date: Wed, 03 May 2006 09:40:43 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories Subject: Re: dualities Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: O X-Status: X-Keywords: X-UID: 167 John Baez wrote: > [...] > So, can we find equally nice examples [of representable dualities] where C and D are instead > 2-categories? In particular, can we find examples where C and D > are 2-categorical generalizations of the 1-categorical examples > we already know? > > In particular, he suggested taking the example where C is the > category of finite distributive lattices and finding an analogous > example where C is the 2-category of (maybe finite, in some sense?) > distributive categories. Enrico Vitale just sent me the answer for that one: C = the 2-category of idempotent-closed categories, D = the 2-category of presheaf categories. This categorifies C = Pos, D = StoneDLat by passing from 2 to Set as the enriching autonomous category (so in that sense one could say we were in 3-CAT all along, though presumably only trivially so by virtue of only having identity modifications when V = 2, I think). Although I'd heard the phrase "Morita equivalence" many times over the years, it meant nothing to me until recently when Bill Lawvere was talking about graphs as presheaves on the monoid consisting of the three monotone functions on the ordinal 2 and I finally woke up to the connection between splitting the two idempotents and ME (the equivalence, not the condition). The idempotent closure of that monoid, meaning the result of splitting the idempotents, is just the initial segment of Delta of length 2, aka the ordinals 1 and 2 and their monotone functions. The impact on the models, here graphs, is that splitting the idempotents results in giving the self-loops that were playing the role of vertices their own datatype V, as coded by the ordinal 1. This new category of graphs is not the old one as its objects now have vertices in their own right, but it is equivalent to the old one. {2} and {1,2}, each made a category with respectively 3 and 7 monotone functions, are Morita equivalent: they have equivalent idempotent closures, and homming into Set maps them to equivalent categories, the iff that makes Morita equivalence important. ME is the kernel of idempotent closure, which is a categorification, with Set in place of 2, of the functor Ord --> Pos (Ord the category of preordered sets, Pos of posets) that collapses the cliques. The reason there is no representable duality between Ord and a suitable cousin of StoneDLat (FinOrd and FinDLat for the Stonaphobes) is that preorders are equivalent to posets and the Yoneda embedding taking elements of P to primes in 2^P, while fully faithful, is only good up to equivalence. (The homfunctor being transposed here is the order <= : P\op x P --> 2.) The categorification of this, meaning in this case not the passage from 2-CAT to 3-CAT but from enrichment in 2 to enrichment in Set, still has to deal with equivalence in the same way (though here it goes with the territory and so is less noticeable than back down at Ord vs. Pos where we tend to think isomorphism rather than equivalence). But Hom: C\op x C --> Set is not itself an equivalence but only a "retract that retracts retracts", the essence of Morita equivalence (a dual of Freyd's "trivial for a trivial reason"?). In order to take the "log to the base Set" we can't really "retract all the retracts" because we may need to keep some of them around but then which ones (like picking a dense subset of a continuum: which subset?). We can however put them all in, which is to say, split all the idempotents, so we do that in order to get a normal form. The rest of this duality is then the triviality that the internal hom of CAT is contravariant in its first argument. Morita equivalence is the only thing to be worried about. Proposition 5.28 of Kelly's "Basic Concepts of Enriched Category Theory", namely that Cauchy completion (Kelly's name for the enriched counterpart of idempotent closure) permits taking logs to any autonomous base V, then produces a proper class of dualities, one for every autonomous V. In particular we can recover Pos\op ~ StoneDLat by taking V = 2. (Pos and Ord, preordered sets, while not equivalent any more than CAT and its subcategory of idempotent-closed categories are equivalent, have equivalent objects which is all we need ask of a duality.) There are two "good" 3-object V's, the non-Heyting one of which enriches the "prossets" that Haim Gaifman and I wrote about in LICS'87, so these have their dual objects in the same way, by homming into 3, a construct I talked about incomprehensibly at the Newton Institute meeting on geometry in computation some years ago, not recognizing that it was a duality. Metric spaces, another duality there. And so on. But then every such duality has its subdualities, for example Set\op ~ CABA as a subduality of Pos\op ~ StoneDLat, so a great many more dualities there. Enrico also mentioned the Gabriel-Ulmer duality for locally finitely presentable categories, and the Adamek-Lawvere-Rosicky duality for varieties. Are these in addition to the above or can they be recovered from them? Likewise for the duality Peter Johnstone mentioned? Vaughan Pratt From rrosebru@mta.ca Wed May 3 17:44:16 2006 -0300 Date: Wed, 3 May 2006 14:45:27 -0400 (EDT) From: Michael Barr To: Categories list Subject: 2-Chu MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: X-Keywords: X-UID: 168 I thought about what John Baez said. No, I had nothing in mind for a 2-Chu, but now I do. First, let me mention that Juergen Koslowski gave a talk on 2-Chu at a meeting in Santa Barbara several years ago. Anyway, what follows isn't a theory, only an example, not worked out in detail. Let us form what I will call Chu(Cat,Set) without getting involved in size issues. (It could, for example, be categories with finite homsets and finite sets.) An object would be (\C,\D,T) where \C and \D are categories and T: \C x \D --> Set is a functor. (I thought of making it \C^o x \D, but in the made this choice. It could be changed.) Anyway, a map in this category is (F,G,\alpha): (\C,\D,T) --> (\C',\D',T') where F:\C --> \C', G: \D' --> \D are functors and \alpha: T(-,G-) --> T'(F-,-) is a natural equivalence. Make the class of maps (\C,\D,T) --> (\C',\D',T') into a category [(\C,\D,T),(\C',\D',T')] with objects as above. A morphism (F,G,\alpha) --> (F',G',\alpha') is (\beta,\gamma), where \beta: F --> F' and \gamma: G --> G' are natural transformations such that \alpha(C,D') T(C,GD') -------------> T'(FC,D') | | | | T(C,\gamma D') | | T(\beta C,D') | | v v T(C,G'D') ------------> T'(F'C,D') \alpha'(C,D') commutes. Now define (\C,\D,T) -o (\C',\D',T') = ([(\C,\D,T),(\C',\D',T')],\C x \D',S) with S either of the isomorphic pairings that takes (F,G,\alpha)(C,D') to T(C,GD') or to T'(FC,D'). Of course, they are isomorphic by \alpha(C,D'), but you still have to choose one or the other. Of course, there is also a choice involved in the direction of \alpha. The dual of an object interchanges the categories and replaces \alpha by its inverse. But you do need an isomorphism here, in order to reverse things. From rrosebru@mta.ca Thu May 4 17:35:04 2006 -0300 Date: Wed, 03 May 2006 23:39:39 -0700 From: Vaughan Pratt MIME-Version: 1.0 To: categories Subject: Re: dualities Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: 7bit Status: O X-Status: X-Keywords: X-UID: 212 Sorry, I should have said "Lawvere's name for..." in > Proposition 5.28 of Kelly's "Basic Concepts of Enriched Category > Theory", namely that Cauchy completion (Kelly's name for the enriched > counterpart of idempotent closure) Vaughan From rrosebru@mta.ca Thu May 4 17:36:33 2006 -0300 From: "Reinhard Boerger" To: categories@mta.ca Date: Thu, 04 May 2006 14:13:59 +0200 MIME-Version: 1.0 Subject: Choice and algebraic closure Content-type: text/plain; charset=US-ASCII Content-transfer-encoding: 7BIT Content-description: Mail message body Status: O X-Status: X-Keywords: X-UID: 213 Hallo, in the PS to his mail on duality, John Baez wrote: > Briefly, while the existence of an algebraic closure of Q > can be shown without choice, it uniqueness-up-to-isomorphism > seems to require choice. Also, while arithmetic operations > in Qbar are computable, they seem to present interesting challenges. it would look interesting to me to show that uniqueness up to isomorphism requires some kind of choice, e.g. by proving that it implies some choice principle. I started thinking about this and came to the following observation: Let K be the field obtained from Q by adjoining all square roots of (positive) primes or - equivalently - of all positive rational numbers. Then an algebraic closureof K is obviously the same as an algebraic closure of Q - even without choice. Now consider an arbitrary sequence of two element sets S(n), w.l.o.g. pairwise disjoint, and let S be their union. Now consider the polynomial ring Q(S) over Q in variables x(s) for all s in S. For each natural number n let p(n) be the n-th prime and consider the poynomials f_n:=x(s)+x(t) and g_n:=x(s)x(t)+p(n) in Q(S) where S(n) consists of the two elements s and t. Now let K' be the factor ring obtained from Q(S) by dividing out all f_n and g_n. If we have a choice function which assigns to each n an element s(n) in S(N), then there is a unique isomorphism from K' to K that maps each s(n) to the (positive) square root of p(n). Conversely, if we have an arbitrary isomorphism j from K' to K, we get a choice function which chooses for the each n unique s(n) in S(n) with j(n)>0. Thus existence of an isomorphism is equivalent to existence of a choice function. If the existence of an algebraic closure of K' could be shown without choice, then an isomorphism fom this algebraic closure to the set of algebraic complex numbers would restrict to an isomorphism between K and K' and thus render a chioce function for the S(n). Thus the uniqueness-up-to-isomorphism would imply choice for countable families of two-element sets. Maybe refinement oft his argument could even be used to get stronger choice principles. Greetings Reinhard Boerger From rrosebru@mta.ca Thu May 4 17:37:53 2006 -0300 Date: Thu, 4 May 2006 11:22:10 -0400 (EDT) From: Michael Barr To: Categories list Subject: Construction of a real closure MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: X-Keywords: X-UID: 214 Is there a reference for the fact that a countable decidably ordered field has a constructable (and decidably ordered) real closure? Maybe 20 years ago, an undergraduate did a project under my supervision proving exactly that (although the proofs are not constructive). While I don't think it would prove feasible for actual computation, the fact that it exists is interesting. Finally, I have looked more closely at it and if it is not known, I think it publishable. Unfortunately, the student (who was a Commonwealth Fellow at Cambridge) lost interest in math and is doing other things. But I think I know how to get in touch with him. Michael From rrosebru@mta.ca Fri May 5 09:31:07 2006 -0300 From: "Marta Bunge" To: mbarr@math.mcgill.ca, categories@mta.ca Subject: RE: Construction of a real closure Date: Thu, 04 May 2006 18:20:08 -0400 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Status: O X-Status: X-Keywords: X-UID: 215 Dear Michael, >Is there a reference for the fact that a countable decidably ordered field >has a constructable (and decidably ordered) real closure? > In my paper "Sheaves and Prime Model Extensions", J. of Algebra 68 (1981) 79-96, there is a proof of the existence of the real closure of an ordered field in any elementary topos, plus considerations about the failure of the existence of the algebraic closure. The context is more general (model theory in toposes), and there are other instances which I do not recall offhand. Maybe that is not what you are asking? I thought that I would mention it, just in case. Best, Marta From rrosebru@mta.ca Fri May 5 09:32:14 2006 -0300 Date: Thu, 4 May 2006 19:28:34 -0400 (EDT) From: Michael Barr To: categories@mta.ca Subject: RE: Construction of a real closure MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: X-Keywords: X-UID: 216 I don't think this is what I was asking, although it may be related. Certainly, the context I have is one in which if you have a real closure and adjoin i, you get an algebraic closure. And the real closure is unique because it is the usual real closure. In fact, the usual real closure is used to prove some things. As I said, the construction is constructive; the proofs are classical. Let me give an example of the flavor. Suppose you want to have a square root of a > 0. It is decidable (by hypothesis) if a > 0; what may not be decidable is whether a has a square root. What the student did (he credits Joyal with some of the main ideas, BTW, and this may be one of them) was to form F[x]/I where I is the ideal of all polynomials that vanish at sqrt(a). Even though it may not be decidable if sqrt(a) in F, it is still decidable if a polynomial vanishes there. First use Sturm's criterion to find an interval that contains exactly one root of f(x) = x^2 - a (in a real closure) and then for any polynomial g, g is in I iff gcd(f,g) has a root in that interval, again using Sturm's criterion. Clearly, F[x]/I contains a square root of a, even if it is not decidable whether that field is F. Michael On Thu, 4 May 2006, Marta Bunge wrote: > > Dear Michael, > > >Is there a reference for the fact that a countable decidably ordered field > >has a constructable (and decidably ordered) real closure? > > > > In my paper "Sheaves and Prime Model Extensions", J. of Algebra 68 (1981) > 79-96, there is a proof of the existence of the real closure of an ordered > field in any elementary topos, plus considerations about the failure of the > existence of the algebraic closure. The context is more general (model > theory in toposes), and there are other instances which I do not recall > offhand. Maybe that is not what you are asking? I thought that I would > mention it, just in case. > > Best, > Marta > > From rrosebru@mta.ca Fri May 5 09:32:53 2006 -0300 From: "Marta Bunge" To: categories@mta.ca Subject: RE: Construction of a real closure Date: Thu, 04 May 2006 18:20:08 -0400 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Status: O X-Status: X-Keywords: X-UID: 217 Dear Michael, >Is there a reference for the fact that a countable decidably ordered field >has a constructable (and decidably ordered) real closure? > In my paper "Sheaves and Prime Model Extensions", J. of Algebra 68 (1981) 79-96, there is a proof of the existence of the real closure of an ordered field in any elementary topos, plus considerations about the failure of the existence of the algebraic closure. The context is more general (model theory in toposes), and there are other instances which I do not recall offhand. Maybe that is not what you are asking? I thought that I would mention it, just in case. Best, Marta From rrosebru@mta.ca Fri May 5 09:33:32 2006 -0300 Date: Thu, 4 May 2006 22:14:02 -0400 (EDT) From: Phil Scott To: Categories list Subject: Re: Construction of a real closure MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: X-Keywords: X-UID: 218 Dear Mike: One reference is the work of M-F Coste-Roy and H. Lombardi in Real Algebraic Geometry; a quick Google search yielded: http://hlombardi.free.fr/publis/AMega90-1.html Also they have some papers in the J.ACM, apparently. In fact, it seems to be a huge area of model theory: see http://name.math.univ-rennes1.fr/michel.coste/Borel/w1abs.html Cheers, Phil On Thu, 4 May 2006, Michael Barr wrote: > Is there a reference for the fact that a countable decidably ordered field > has a constructable (and decidably ordered) real closure? > > Maybe 20 years ago, an undergraduate did a project under my supervision > proving exactly that (although the proofs are not constructive). While I > don't think it would prove feasible for actual computation, the fact that > it exists is interesting. Finally, I have looked more closely at it and > if it is not known, I think it publishable. Unfortunately, the student > (who was a Commonwealth Fellow at Cambridge) lost interest in math and is > doing other things. But I think I know how to get in touch with him. > > Michael > > > > From rrosebru@mta.ca Fri May 5 09:34:14 2006 -0300 Date: Fri, 5 May 2006 13:58:36 +0430 From: "Darush Aghababaee" To: categories@mta.ca Subject: algebraic closure MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Status: O X-Status: X-Keywords: X-UID: 219 dear all i think if Q has discrete topology then algebraic closure of Q as same as topological closure Q so that study of cl(Q) is easier. is it true? From rrosebru@mta.ca Fri May 5 09:35:00 2006 -0300 Date: Fri, 5 May 2006 08:22:58 -0400 (EDT) From: Michael Barr To: Categories list Subject: Re: Construction of a real closure MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: X-Keywords: X-UID: 220 Yes, that seems exactly the sort of thing I was asking about. Michael On Thu, 4 May 2006, Phil Scott wrote: > Dear Mike: > One reference is the work of M-F Coste-Roy and H. Lombardi in Real > Algebraic Geometry; a quick Google search yielded: > > http://hlombardi.free.fr/publis/AMega90-1.html > > Also they have some papers in the J.ACM, apparently. > > In fact, it seems to be a huge area of model theory: see > http://name.math.univ-rennes1.fr/michel.coste/Borel/w1abs.html > > Cheers, > Phil > From rrosebru@mta.ca Fri May 5 16:01:02 2006 -0300 From: "Marta Bunge" To: categories@mta.ca Subject: RE: Construction of a real closure Date: Fri, 05 May 2006 09:30:25 -0400 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Status: O X-Status: X-Keywords: X-UID: 221 Dear Michael, It is related to what I did, but in a different way. BTW, I remembered something incorrectly. In my J. Alg '81 paper, I actually use (not prove) Joyal's result on the existence of the real closure of an ordered field ("Cloture algebrique reelle d'un corps ordonne", Lecture, Universite de Montreal, January 1979). The proof is indeed classical, so this holds (at least) in any topos which is a BVMST (Boolean-valued-model of Set Theory), not in every topos. I show that for certain toposes E, the result is till true. I was mostly interested in toposes E which arise in sheaf represenation theorems for algebraic structures (such as von Neuman regular (differential) rings). First, I prove a general theorem for quotients T-->T* of coherent theories satisfying (what I call) the "Sturm property" in a topos E, to the effect that prime model extensions of Mod_{E}(T) in Mod_{E}(T*) exist and are preserved by continuous functors. For instance, I show that the real closure exists in toposes E = Sh_fc(B) (sheaves for finite covers in a Boolean algebra B). The example of T--->T* with T = {ordered fields} and T* = {real closed fields} is shown to have the Sturm property in E = Sh_{fc}(B) by resorting to a "mid-way-house method", first via the Gleason cover of E, and then taking its topos of double negation sheaves, which is a BVMST. By transfer, I obtain that every commutative regular f-ring (in Sets) has an (invariant, and an atomless) real closure. Another example is that of the differential closure of a differential field and a transfer to differential Von Neuman regular rings of non-zero characteristic. The question of characterizing toposes E for which a certain quotient T-->T* of coherent theories has the "Sturm property" is open. As I said, I was only interested in sheaf representations by global sections, so I only looked at such toposes. Best, Marta >I don't think this is what I was asking, although it may be related. >Certainly, the context I have is one in which if you have a real closure >and adjoin i, you get an algebraic closure. And the real closure is >unique because it is the usual real closure. In fact, the usual real >closure is used to prove some things. As I said, the construction is >constructive; the proofs are classical. > >Let me give an example of the flavor. Suppose you want to have a square >root of a > 0. It is decidable (by hypothesis) if a > 0; what may not be >decidable is whether a has a square root. What the student did (he >credits Joyal with some of the main ideas, BTW, and this may be one of >them) was to form F[x]/I where I is the ideal of all polynomials that >vanish at sqrt(a). Even though it may not be decidable if sqrt(a) in F, >it is still decidable if a polynomial vanishes there. First use Sturm's >criterion to find an interval that contains exactly one root of f(x) = x^2 >- a (in a real closure) and then for any polynomial g, g is in I iff >gcd(f,g) has a root in that interval, again using Sturm's criterion. >Clearly, F[x]/I contains a square root of a, even if it is not decidable >whether that field is F. > >Michael > From rrosebru@mta.ca Fri May 5 16:01:52 2006 -0300 Date: Fri, 5 May 2006 09:45:45 -0400 (EDT) From: Peter Freyd Message-Id: <200605051345.k45DjjeM026892@saul.cis.upenn.edu> To: categories@mta.ca Subject: Re: Construction of a real closure Status: O X-Status: X-Keywords: X-UID: 222 Mike asks: Is there a reference for the fact that a countable decidably ordered field has a constructable (and decidably ordered) real closure? The expert on all such questions is Anil Nerode .See his Effective content of field theory (Ann. Math. Logic 17 (1979), no. 3, 289--320) for a collection of all the relevant results. John writes: Briefly, while the existence of an algebraic closure of Q can be shown without choice, it uniqueness-up-to-isomorphism seems to require choice. Also, while arithmetic operations in Qbar are computable, they seem to present interesting challenges. The need for choice could hardly arise when working with a decidable countable structure such as Q. One way of naming an algebraic real number is with an ordered triple , where l and r are rationals, P a monic polynomial with rational coeficients that is irreducible over the rationals (the decidablity of which can be found in van der Waerden) such that P(l)P(r) < 0 and R(l)R(r) > 0 for R any of the non-tivial iterated derivatives of P. Another such triple names the same element iff the polynomials are the same and the intervals overlap. Effective constructions for the ordered-field operations in this context are pretty standard. From rrosebru@mta.ca Fri May 5 16:02:22 2006 -0300 Date: Fri, 5 May 2006 15:59:26 +0200 (CEST) From: Paul-Andre Mellies To: categories@mta.ca Subject: Spring School in Theoretical Computer Science MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII; format=flowed Status: O X-Status: X-Keywords: X-UID: 223 *** Last call for participation *** Spring School in Theoretical Computer Science EPIT 2006 Games in Semantics and Verification May 29 -- June 2, 2006 Ile de Re, France http://epit.pps.jussieu.fr * * * THE SPRING SCHOOL The Spring School in Theoretical Computer Science is a French institution, started by Maurice Nivat in 1973. The School is based on a simple but extraordinarily successful recipe: bring together the world's specialists of a different topic every year, for a week, in a beautiful and somewhat recluded part of France. This year, the Spring School will meet at <> --- an exquisite island on the Atlantic Ocean, in front of La Rochelle. The seaside setting will certainly provide lots of opportunities for informal discussions and collaborations in the evenings. GAME THEORY IN SEMANTICS AND VERIFICATION The Spring School 2006 is designed for students and researchers interested in learning more about Game Theory and its recent applications to the Semantics and Verification of Programs and Programming Languages. The Spring School offers the first international platform for discussing together the recent advances in these two extremely active topics. The Spring School is openly multi-disciplinary, and will also introduce related topics such as Descriptive Set Theory in Mathematical Logic, or Nash equilibrium in Economic Games. Lectures will be provided ***in english*** by: Samson Abramsky (Oxford, UK) Jacques Duparc (Lausanne, Switzerland) Paul Gastin (LSV, ENS Cachan) Erich Gradel (Aachen, Germany) Martin Hyland (Cambridge, UK) Luke Ong (Oxford, UK) Tristan Tomala (Ceremade, Paris Dauphine) Igor Walukiewicz (Bordeaux) Wieslaw Zielonka (Paris) The program of the Spring School appears at: http://epit.pps.jussieu.fr/ REGISTRATION We have done our best to limit the registration and accomodation fees to the following amounts: --- 250 euros for students in double rooms, --- 500 euros for researchers in single rooms. REGISTRATION We have done our best to limit the registration and accomodation fees to the following amounts: --- 250 euros for students in double rooms, --- 500 euros for researchers in single rooms. This includes the (somptuous) accomodation, the (perfect) meals, the coffee breaks, as well as a coach from La Rochelle to l'Ile de Re, and return. DEADLINES An unexpected increase in our financial support enables us to accept more participants than what was originally forecast. There remains however only a very limited number of places. We thus ask that interested people contact the organizers *directly* and *immediately* at: mellies@pps.jussieu.fr muscholl@liafa.jussieu.fr ... immediately meaning before Wednesday, May the 10th, ideally, and at the latest on Monday, May the 15th. Further information is provided on the web page of the Spring School: http://epit.pps.jussieu.fr/ GIVE A TALK Depending on the time available, it should be possible for participants to give a talk on their work at the Spring School. Please indicate a title and abstract when you register to the School. Looking forward to meeting you at l'Ile de Re, -- The organizers Paul-Andre Mellies (PPS, Paris) and Anca Muscholl (LIAFA, Paris) From rrosebru@mta.ca Fri May 5 16:03:23 2006 -0300 Subject: Re: Construction of a real closure To: categories@mta.ca (categories) Date: Fri, 5 May 2006 07:55:57 -0700 (PDT) From: "John Baez" MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Status: O X-Status: X-Keywords: X-UID: 224 Peter Freyd writes: >John writes: >> Briefly, while the existence of an algebraic closure of Q >> can be shown without choice, it uniqueness-up-to-isomorphism >> seems to require choice. Also, while arithmetic operations >> in Qbar are computable, they seem to present interesting challenges. >The need for choice could hardly arise when working with a decidable >countable structure such as Q. Above I was reporting what David Madore wrote. He indeed claims that in ZF without C one cannot prove the uniqueness of the algebraic closure of Q. He also said some other interesting stuff, so I might as well quote it verbatim: From: david.madore@ens.fr (David Madore) Newsgroups: sci.math.research Subject: Re: The algebraic closure of the rationals Date: Fri, 7 Apr 2006 14:12:15 +0000 (UTC) Organization: Ecole Normale Superieure, Paris John Baez in litteris scripsit: > 1) Is there a way to enumerate the elements of Qbar such that > relative to this enumeration, the field operations are computable? > If so, how efficiently can they be computed? If not, how far > up the hierarchy of impossible-to-actually-compute functions do > we have to go? The usual manner is to represent a real element of Qbar by * its minimal polynomial over Q (or perhaps, some polynomial, not necessarily minimal, but probably at least separable, of which it is a root), * an interval which isolates the root from all other roots (or the number of the root in the usual order on the reals). Basically the trick is that sums and products can be computed by universal rules (if P1 and P2 are polynomials over Q, there is a polynomial, which can be given universally in function of the coefficients of P1 and P2, whose roots are the sums of roots of P1 and P2, and ditto for the product), and roots can always be isolated using Sturm-Liouville (in other words, you can narrow the interval as much as you want since Sturm-Liouville lets you count the number of roots in any given interval). This is for real algebraics; for the full Qbar, you just represent a complex number by its real and imaginary parts (both of which are algebraic if the complex is algebraic). Actually programming this is *unbelievably* painful. As for the algorithmic complexity, I think it's not that bad, in the sense that if x and y have small height (for any reasonable definition of "height") then computing x+y can be done in a reasonable time, but there's a catch: the height of x+y grows considerably larger than that of x or y, so any actual computation can become terribly difficult. (The same problem happens for rationals: computing r+s where r and s are rationals is polynomial in the height of r and s, but try computing something like 1/2+1/3+1/5+1/7+1/11+1/13+1/17...) For some specific uses it is better to use the p-adics than the reals: the advantage is that p-adic approximation is *much* easier than real approximation (thanks to the ultrametric properties); the disadvantage is that whereas the reals are "almost" algebraically closed, the p-adics are quite far from it, and representing elements of the algebraic closure of Q_p is again quite a nuisance. > 2) On the other extreme: can we even prove the existence of Qbar > without the axiom of choice, or perhaps countable choice? Yes, the *existence* of Qbar, or of any countable or even well-orderable field (and various others, such as Q_p), can be shown without the axiom of choice. However, the *uniqueness* of Qbar requires the axiom of choice: it is consistent in ZF alone that there exists an uncountable algebraic closure of Q. (Basically it's true in ZFA because you can create atoms for elements of Qbar and let Galois act on them and take the usual permutation model, and then some embedding theorem gives the result in ZF.) I'm not sure about whether one needs AC to show that the algebraic closure of Q constructed using the reals and the p-adics as explained above actually give the same thing. > 3) I've heard that it's even hard to get an "explicit" description > of the algebraic closures of finite fields - are there any > theorems to this effect? I don't think it's hard. In fact, here's a very elegant and intriguing explicit construction of the algebraic closure of F_2, which is due to Conway: Define two binary operations, '#' and '@', on the class of ordinals, by transfinite induction, by letting: x # y = mex ( { x'#y | x' To: baez@math.ucr.edu, categories@mta.ca.barr.anil@math.cornell.edu Subject: Re: Construction of a real closure Status: O X-Status: X-Keywords: X-UID: 225 If memory serves me, Nerode proved years ago that the necessary and sufficient condition that all the algebraic closures of a field are effectively isomorphic is that its polynomials effectively factor into irreducibles. As for the later, the ancient theorem is that if a unique factorization domain has only finitely many units and if the factorization is effective then so it is for its polynomial ring. The condition on units pretty much forces the proof. From rrosebru@mta.ca Sat May 6 11:46:52 2006 -0300 Subject: Construction of a real closure From: Eduardo Dubuc To: categories@mta.ca Date: Fri, 5 May 2006 23:24:38 -0300 (ART) MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Status: O X-Status: X-Keywords: X-UID: 226 Hi! In 1978 Joyal gave a lecture in Montreal where he described a constructive construction of the pithagorean closure (every a > 0 has a square root > 0). This he does with a beautifully simple idea. Assume K is ordered decidable (in particular "=" is decidable) Using M. Barr wording: " Let me give an example of the flavor. Suppose you want to have a square root of a > 0. It is decidable (by hypothesis) if a > 0; what may not be decidable is whether a has a square root." Joyal simply construct the ring A = K[x]/(x^2 - a) = K[s] with s^2 = a. He then considers an equivalence relation in A: Consider J = {x + ys | x = 0, y = 0 or not(y = 0), -x/y > 0, (-x/y)^2 = a} the relation "\in J" is decidable, then if J is an ideal, the quotient by J will also have "=" decidable. J = {0} precisely if a does not have an square root in K, but it can be shown that J is an ideal without having to decide if a has or not a square root !! Furthermore, it can be shown (with some but not too much work) that the quotient K[s]/J is a order decidable field. This field is the solution to the universal problem of adding to K an square root of a, and it is obtained without the use of Gauss theorem !! (Notice that for non-ordered fields, the universal solution of adding a square root of a when a already has a square root DOES NOT EXIST (it is not given by K itself), while for ordered fields the solution of adding a positive square root does exist, and it is K itself) Algorithms for K[s]/J are easily obtained from the algorhitms for K !! We can iterate this, and then construct the filtered colimit. This filtered colimit is the pithagorean closure, and by construction of filtered colimits a rather simple description of its elements is there. Also algorithms for this colimit are rather easily obtained. All this is considerably simpler than other methods quoted in this thread, and certainly not " *unbelievably* painful". Actually, I believe it could be possible to write with reaonable trouble an actual computer program to calculate in the pithagorean closure of the rationals. I have an article written with an student for a journal on math teaching ("revista de educacion matematica" of the UMA, Union Matematica Argentina) with details of all this. The text is in spanish, in a .pdf file, and I will be happy to send it on request I do not know if Joyal had also a continuation of this story, meaning how to keep adding more roots without Gauss's theorem. e.d. From rrosebru@mta.ca Sat May 6 11:48:02 2006 -0300 Date: Sat, 6 May 2006 13:39:42 +0930 From: David Roberts To: categories@mta.ca Subject: gr-stacks (revised) MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1 Content-Transfer-Encoding: 8bit Status: O X-Status: X-Keywords: X-UID: 227 Categorists, On Sat, Apr 29, 2006 at 01:27:20PM +0930, David Roberts wrote: > Dear all, > > after a bit of searching, I cannot find much in the literature about gr- > stacks, more specifically, charts and presentations thereof reflecting (in a > non-technical sense) the group-like structure. Also, aside from self > equivalences of gerbes and quotients of groups **(G/H for non-normal H)**, I > cannot dream up other "interesting" examples - and these are the opposite ends > of the spectrum I want to consider. A bit of confusion occured when I tried to post a corrected version of the above (all due to myself), so here goes. I retract the statement in ** ** above - what I meant was G/H with a badly behaved topological/differentiable quotient (H normal in G) and I was after examples not connected with gerbes/crossed modules but something more interesting' than group quotients. Thanks, -- David Roberts Pure Mathematics University of Adelaide South Australia, 5005 You know we all became mathematicians for the same reason: we were lazy. -Max Rosenlicht(1949) From rrosebru@mta.ca Sat May 6 11:49:07 2006 -0300 Date: Sat, 6 May 2006 09:53:49 -0400 (EDT) From: Michael Barr To: categories Subject: Re: Construction of a real closure MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: X-Keywords: X-UID: 228 John's post was interesting, but a couple of things bother me. Since there are explicit enumerations of Q[x] and you can order the roots of any polynomial by their order and adjoin the roots in order, how can you get an uncountable closure? Another question (for Q, although any Archimedean field should work) what if you use Cauchy sequences generated by Newton iteration, having isolated the real roots using Sturm and sticking to an interval in which the derivative is positive (I am thinking of a monic polynomial that is increasing in the end)? Incidentally, AC doesn't bother me at all, but constructive methods are interesting in their own right. Michael From rrosebru@mta.ca Sun May 7 17:08:54 2006 -0300 From: "Ronnie Brown" To: Subject: Remark on: Re: categories: gr-stacks (revised) Date: Sat, 6 May 2006 23:13:57 +0100 MIME-Version: 1.0 Content-Type: text/plain;format=flowed; charset="iso-8859-1";reply-type=original Content-Transfer-Encoding: 7bit Status: O X-Status: X-Keywords: X-UID: 229 David, When you have bad' quotients you can look at the equivalence relation, a special case of a groupoid. In the case of subgroups H < G, then you also get a covering groupoid G' \to G with vertex groups isomorphic to H. If you want more than one subgroup, you might need global actions or groupoid atlases http://www.informatics.bangor.ac.uk/public/mathematics/research/preprints/06/algtop06.html#06.02 Perhaps you can find bad' quotients which nonetheless have good' holonomy groupoids, analogously to the foliation case? That would be good, but is it too naive an idea? On the other hand, an irrational flow on a torus gives a foliation of the torus which has a smooth holonomy groupoid. see preprint 06.03 and the information there for some ideas on holonomy. Ronnie Brown ----- Original Message ----- From: "David Roberts" To: Sent: Saturday, May 06, 2006 5:09 AM Subject: categories: gr-stacks (revised) > Categorists, > > > On Sat, Apr 29, 2006 at 01:27:20PM +0930, David Roberts wrote: > >> Dear all, >> >> after a bit of searching, I cannot find much in the literature about gr- >> stacks, more specifically, charts and presentations thereof reflecting >> (in a >> non-technical sense) the group-like structure. Also, aside from self >> equivalences of gerbes and quotients of groups **(G/H for non-normal >> H)**, I >> cannot dream up other "interesting" examples - and these are the opposite > ends >> of the spectrum I want to consider. > > > A bit of confusion occured when I tried to post a corrected version of the > above (all due to myself), so here goes. > > I retract the statement in ** ** above - what I meant was G/H with a badly > behaved topological/differentiable quotient (H normal in G) and I was > after > examples not connected with gerbes/crossed modules but something more > interesting' than group quotients. > > Thanks, > > -- > David Roberts > Pure Mathematics > University of Adelaide > South Australia, 5005 > > You know we all became mathematicians for the same reason: we were lazy. > -Max Rosenlicht(1949) > > > From rrosebru@mta.ca Mon May 8 10:15:33 2006 -0300 From: "Marta Bunge" To: categories@mta.ca Subject: RE: Construction of a real closure Date: Sun, 07 May 2006 16:28:38 -0400 Mime-Version: 1.0 Content-Type: text/plain; format=flowed Status: O X-Status: X-Keywords: X-UID: 230 Dear Michael, >Incidentally, AC doesn't bother me at all, but constructive methods are >interesting in their own right. Constructive methods are sometimes more than "just interesting in their own right". If a certain theorem expressible in geometric logic can be proved constructibly, then it is true in any topos, and so amenable to a variety of interesting interpretations. I return to one of the examples I mentioned from my paper ("Sheaves and prime model extensions", J. Algebra 68, 1981) to illustrate how it works for the real closure, which has both a constructive and a non-constructive part. Preamble. The following theorem has been shown (Lipshitz, Trans AMS 211, 1975; van der Dries Ann. Math. 12, 1977) in order to solve the analogue of Hilbert's 17th problem for commutative (von Neumann) regular f-rings. Theorem. Any commutative regular f-ring has an atomless real closure. Using the Pierce representation of any commutative (von Neumann) regular f-ring R by an ordered field K in the (non-Boolean) topos E of sheaves on the spectrum of K, the above theorem can be shown more easily than in the above quoted sources, simply using the analogue for ordered fields, with some care, since we know that, although there is an algorithm for constructing the "real closure" of an ordered field, this does not automatically give that such is real closed. This is how I proceed, using the "mid-way house" method. The inclusion F = E_{not not} >---> E factors through the Gleason cover g:G--->E (g a surjection) via a flat inclusion i: F >---> G. Since F is a BVM/ST, i^*g^*K >---> K' has a real closure. Thus, since i_* preserves finitary logic, g^*K >---> i_*K' is an extension of g^*K into a real closed field i_*K'. Sturm's theorem gives an algorithm which makes sense in any topos, provided the ordered field one applies it to is already contained in some real closed field. From this follows first that g^*K has a real closure in G, and then, by the surjectivity of g (g^* faithful), follows that K has a real closure in E. By "transfer", the commutative regular f-ring R ( R = \Gamma (K)) has a real closure (in Sets). Other results (including classically new ones) in the case of differential rings are proved in that paper. Sheaf methods in the theory of commutative rings are very useful in classical mathematics. See for instance "Recent Advances in the Representation Theory of Rings and C*-Algebras by Continuous Sections", Memoirs AMS 148, 1973. In particular, C. Mulvey, "Intuitionistic Algebra and Representation of Rings" in that volume). Another source is "Applications of Sheaves" (Proceedings Durham 1977), LNM 753, Springer 1977. In particular, the papers by Fourman and Hyland, Fourman and D. Scott, and C. Rousseau. Anyway, this is a huge subject and hints at the usefulness of constructive methods in algebra and analysis, when available. Anhyway, trhis is old stuff. Best, Marta From rrosebru@mta.ca Mon May 8 10:18:41 2006 -0300 From: "Reinhard Boerger" Organization: FernUniversitaet Date: Mon, 08 May 2006 12:15:25 +0200 MIME-Version: 1.0 Subject: Re: Choice and algebraic closure To: categories@mta.ca Content-type: text/plain; charset=ISO-8859-1 Content-transfer-encoding: Quoted-printable Content-description: Mail message body Status: O X-Status: X-Keywords: X-UID: 255 Andrej Bauer wrote: > Reinhard Boerger wrote: > > For each natural number n let p(n) be the n-th prime and > > consider the poynomials f_n:=3Dx(s)+x(t) and g_n:=3Dx(s)x(t)+p(n) in = Q(S) > > What precisely is the status of the expressions "x(s)+x(t)" and > "x(s)x(t)" here? It is not clear to me that, given a two-element set > which possibly cannot be ordered, we can form a sum or a product indexed > by the set. I am afraid you're covertly ordering the two element set > when you name its elements s and t, then form the sum x(s)+x(t). After > all, a polynomial in Q[S] is going to be an (equivalence class of) > sequence of pairs (coefficient, monomial). So if we could form x(s)+x(t) > we could (perhaps) also order s and t by looking at the sequence which > represents x(s)+x(t). Am I raising any doubts in your mind? I think some > details need to be worked out here. You are right; without choice we should be careful, and several classicall= y equal things become distinct. I am used to the following way of defining the pol= ynomial ring over a ring A in variables x(s) for all s from a given set S: Monomials ar= e maps from S to the set of nonnegative integers, which map all but finitely many elem= ents to 0. The images under this map are considered as exponents, and monomials are multiplied by adding the exponents, and x(s) is the monomial that maps s t= o 1 and all other elements to 0. Then x(s)x(t) is literally the same as x(t)x(s). Anal= ogously, we can define polynomials as maps from the set of all monomials to A, which a= re 0 almost every where; the images are interpreted as coefficients. Then we al= so have x(s)+x(t)=3Dx(t)+x(s). However, also another definition should turm the polynomial ring into a co= mmutative rings; so the elements could also be defined as equivalence classes of (fi= rst-order) terms. The point is that x(s)x(t) should bet taken as an element of the (c= ommutative) polynomial ring, not as a term; the terms x(s)x(t)and x(t)x(s) are indeed = different. The algebraic closure of Q (or of a finite field) can be constructed by su= ccessively adding zeros of polynomials. Since the set of all polynomials can be numbe= red, an isomorphism between two different algebraic closures can be found, if on e= ach step we find a bijection between the finte sets of zeros of a given polynomials= in both closures. This choice usually depends on the privious choices; so countabl= e dependent choice in finite sets (or K=F6nig's Lemma) should render the des= ired isomorphism. On the other hand, the situation is very specific, I do not s= ee how K=F6nig's Lemma should follow from the existence of isomorphism between al= gebraic closures of Q. So I considered an easier case: The choices of square roots of different p= rimes are independent. So the isomorphism between K and K' in my example can be obta= ined uniquely if for every prime you have bijectin between the set of the two s= quare roots of a prime p in K and the set of the square roots of p in K'. ButK is orde= red by construction; so you obtain the bijection if fir each p you choose which o= ne of the two square roots of p shall be mapped to the positive square root in K': If yo= u have the isomorphism, you get an ordering of K' and therefore a choice of positive= square roots. So had to build the field K' for a given sequence of two-element se= ts without using a choice for these sets. Of course the result is not satifactory; it was just my first idea when I = read John Baez' mail. Since there are also irreducible polynomials of degrees >2, I think = that the sock axiom (i.e. countable choise for sequences of two-element sets, i.e. choos= ing a left one for countably many pairs of symmetric socks) should not imply the uniq= ueness of the algebraic closure of Q up to isomorphism. In order to getthe converse,= we should need a way to extend an isomporphism between K and K' to an isomorphism between algebraic closures without using choice again. = Greetings = Reinhard From rrosebru@mta.ca Mon May 8 21:47:38 2006 -0300 Mime-Version: 1.0 (Apple Message framework v749.3) Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: categories@mta.ca From: Thorsten Altenkirch Subject: strict symmetric monoidal Date: Mon, 8 May 2006 23:29:36 +0100 Status: O X-Status: X-Keywords: X-UID: 256 Hi everybody, I haven't been reading this list for a while but now I have a question people on categories may have thought about. May be trivial... apologies in advance. A strict monoidal category is a monoidal one where all the isos are indentities, e.g. we have that A(x)(B(x)C) = (A(x)B)(x)C and also f(x)(g(x)h) = (f(x)g)(x)H because the natural iso is the identity. A nice example is the skeletal cat of finite sets, i.e. objects are natural numbers and Hom m n = {i {i Message-Id: <200605091426.k49EQlPe026995@saul.cis.upenn.edu> To: categories@mta.ca Subject: Whatever happened to Doug Gurr? Status: O X-Status: X-Keywords: X-UID: 257 [Doug Gurr published 7 papers in the 90s with Carolyn Brown on categories, allegories, quandles, and such.] Copyright 2006 Telegraph Group Limited The Daily Telegraph (LONDON) May 9, 2006 Tuesday SECTION: CITY; BUSINESS CLUB; Pg. 9 LENGTH: 1185 words HEADLINE: money case study Uncaging big opportunities in corner shops A just-in-time delivery service for convenience stores has hit profitability at a pounds 133m turnover. So what next, asks Philip Smith? BYLINE: PHILIP SMITH BODY: ARMED with a doctorate in mathematical logic and theoretical computing, backed by years of experience as a management consultant in the retail sector, Douglas Gurr set out to devise a system that would allow convenience stores to benefit from the same buying power, stock rotation and low pricing as the major supermarkets. He found the right formula. His Blueheath Holdings has grown from zero to a pounds 133m turnover business in six years. The challenge for Dr Gurr, 41, is how to maintain that growth and reach his target of pounds 1bn. He also needs to consider whether the Blueheath blueprint can be replicated across other sectors or in other countries. What intrigued Dr Gurr was how small, independent grocery stores got their supplies and, crucially, how they could compete against the big names. "There were two options: either they went to traditional cash-and-carry wholesalers and physically brought back thousand of pounds worth of stock, or they paid extra for a delivery. ''The first ate into their time, the second into the bottom line. We thought we could take enough costs out of the system to be able to offer all the benefits of a delivered service but at a cash-and-carry price,'' he says. Blueheath keeps its costs down by using technology: its system predicts what stock will be wanted by its customers. That, says Dr Gurr, is based on a range of parameters, including fashions, bank holidays, weather and a customer's own stock levels. Blueheath then orders from suppliers in anticipation of the orders that it will get from its array of shopkeeper customers. This allows the company to keep stock levels low but fulfilment levels high. Fulfilment, which runs at around 95pc in the industry, says Dr Gurr, is the level at which a customer's order is actually delivered at the store. The missing products are mainly due to the wholesaler being out of stock. "We operate at 98pc fulfilment,'' he says. "We live or die by our fulfilment and on-time delivery percentages. If you get those right, most of the rest follows.' ' Predictive ordering also means that stock remains with Blueheath for a much shorter time, allowing it to service the whole of the UK from three depots: in east London, the Midlands and Wales. Blueheath has its own fleet of trucks but the bulk of its deliveries are made using spare capacity in established hauliers. That, along with a policy to lease its buildings, means that the Islington-based business is light on capital assets. Blueheath was formed in 2000 as a regional operation, allowing it to develop a model on which it could raise funds for a UK rollout. Two rounds of venture capital funding were followed in 2004 by a listing on Aim. The two initial main funders remain principal shareholders. Its slower-than- expected path to profitability hit its share price, which halved to around 58p. This year Dr Gurr expects Blueheath to turn an operational profit for the first time. It is the scalability of the system that allows the business to look at much higher margins in the future. "This is a scalable business,'' he says. "Scale allows you to get better buying terms, operational improvements and, because the majority of costs are fixed, improved profitability. We are now at the stage where we have reached the scale to become profitable. It's taken a while to get here but it doesn't work as a business until you reach the kind of levels we have.'' Blueheath currently has 4,000 regular customers, serviced by 200 staff, mostly in sales and fulfilment. Continued growth is what Blueheath needs. The sector is worth pounds 17.5bn, of which the largest players account for pounds 2bn to pounds 3bn each. "The bigger we can get it, the better it will work. There is no reason why this shouldn't be a pounds 1bn-turnover business,'' he says.'' Growth would be self-funding but Dr Gurr is adamant that the company will not advertise. "We have never spent a penny on advertising and as long as I'm here we never shall. It's not an appropriate use of funds,'' says Dr Gurr. Blueheath Holdings was the overall national winner in last year's DTI/Interforum E-Commerce Awards. Expert view Chris Jackson Head of finance and management faculty, Institute of Chartered Accountants in England & Wales BLUEHEATH'S best growth strategy is market penetration, to grow turnover by finding more of the same. The largest two players control 29pc of the sector, leaving the remainder fragmented among smaller players. Sales effort here should focus on Blueheath's superior fulfilment and better pricing (the larger competitors are likely to be able to beat Blueheath on price through economies of scale). This strategy maximises return from the current infrastructure. It is also a low-risk strategy compared with alternatives such as licensing or entering foreign markets, which would take Blueheath outside its expertise. Word of mouth will not suffice and advertising is probably not the answer. In the business-to-business sector, direct marketing is the solution. There will be data for sale to assist with identifying the best prospects, which the sales force could then target.John Timpson Chairman, Timpson IN ONLY six years Blueheath has acquired 4,000 customers and a pounds l33m turnover, but it hasn't made any money. It is time for Douglas Gurr to stop living the dream of his business plan and face up to reality. Dr Gurr puts his faith in economies of scale but it may not be so easy to find the extra customers that he anticipates. A safer route would be to create profit from the existing business. He needs to increase margins and that probably means putting up prices. If customers feel that the Blueheath service is helping them to make more money they will be willing to pay a proper price. There is more to business than mathematical logic. In today's technological age, wizard computer solutions are taken for granted - it's old fashioned flair and the personal touch that make the difference between success and failure. Ultimately, profits always depend on people. Dr Gurr should visit as many shops as possible and talk to his customers face to face. They will tell him all that he needs to know.Amanda Rendle Head of business marketing, HSBC ADVERTISING can play a role in creating awareness but there are other marketing techniques Blueheath could use to keep its profile high. For example, using customer testimonials and case studies would deliver the controlled growth Douglas Gurr requires. These approaches should be used primarily via the internet. Word of mouth is also highly effective; both customers and sales teams have a role to play here. By offering a reward scheme - money off or something more tangible - many customers will gladly tell their colleagues about Blueheath 's price and service levels. Customers who make recommendations tend to do so many times - it will only take a few endorsements to magnify the Blueheath message. The sales force should also be ambassadors for the company, both by making sales and by talking about the way that Blueheath operates - emphasising the cultural aspects of the business and describing how they can be a benefit From rrosebru@mta.ca Thu May 11 18:15:10 2006 -0300 From: "Ronnie Brown" To: Subject: Topology and groupoids, by Ronald Brown Date: Tue, 9 May 2006 22:00:00 +0100 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Status: O X-Status: X-Keywords: X-UID: 258 This is to report that because I have uploaded a corrected file to the = printers it will take maybe 14 days for the Booksurge site to offer this = book.=20 However I have (the last!) 15 or so copies of the first printing = available from me at =A313 + P&P and I expect UK printed copies of the = new version fairly soon at =A317 +P&P. More details at: www.bangor.ac.uk/r.brown/topgpds.html From rrosebru@mta.ca Thu May 11 18:15:47 2006 -0300 Date: Wed, 10 May 2006 16:10:48 +0200 From: Josep Maria Font Subject: Master in Pure and Applied Logic Message-id: <9d86514c905fc9ff8fff2fe76a5cfbe3@ub.edu> MIME-version: 1.0 (Apple Message framework v623) Content-type: text/plain; format=flowed; charset=US-ASCII Content-transfer-encoding: 7bit Status: O X-Status: X-Keywords: X-UID: 259 The University of Barcelona and the Technical University of Catalonia (Barcelona) jointly organize a MASTER IN PURE AND APPLIED LOGIC aimed at giving graduate students a thorough grounding in all aspects of advanced logic, both pure and applied. On completion, students will have the necessary skills to be able to continue with their postgraduate studies, put their knowledge into practice in the job market, or start undertaking research in many of the central areas in the field of logic. For more information visit ***** ***** ***** ***** ***** ***** ***** From rrosebru@mta.ca Thu May 11 18:16:18 2006 -0300 Date: Thu, 11 May 2006 12:23:06 +0200 From: Carlos Areces User-Agent: Thunderbird 1.5.0.2 (X11/20060420) MIME-Version: 1.0 To: Carlos.Areces@loria.fr Subject: CFP: International Workshop on Hybrid Logic 2006 (HyLo 2006) Content-Type: text/plain; charset=3DISO-8859-1; format=3Dflowed Content-Transfer-Encoding: 8bit Status: O X-Status: X-Keywords: X-UID: 260 ******************************************************************* FINAL CALL FOR PAPERS International Workshop on Hybrid Logic 2006 (HyLo 2006) Affiliated with LICS 2006 August 11, 2006, Seattle, USA ******************************************************************* AIMS AND SCOPE: Hybrid logic is a branch of modal logic in which it is possible to directly refer to worlds/times/states or whatever the elements of the (Kripke) model are meant to represent. Although they date back to the late 1960s, and have been sporadically investigated ever since, it is only in the 1990s that work on them really got into its stride. It is easy to justify interest in hybrid logic on applied grounds, because of the usefulness of the additional expressive power. For example, when reasoning about time one often wants to build up a series of assertions about what happens at a particular instant,and standard modal formalisms do not allow this. What is less obvious is that the route hybrid logic takes to overcome this problem (the basic mechanism being to add nominals --- atomic symbols true at a unique point --- together with extra modalities to exploit them) often actually improves the behavior of the underlying modal formalism. For example, it becomes far simpler to formulate modal tableau, resolution, and natural deduction in hybrid logic, and completeness and interpolation results can be proved of a generality that is not available in orthodox modal logic. Hybrid logic is now a mature field, therefore a theme of special interest at this HyLo workshop will be the combination of hybrid logic with other logics, the basic methodological question being "what is the best way of hybridizing a given logic?" However, submissions in all areas of hybrid logic are welcome. The workshop HyLo 2006 is likely to be relevant to a wide range of people, including those interested in description logic, feature logic, applied modal logics, temporal logic, and labelled deduction. The workshop continues a series of previous workshops on hybrid logic, for example the LICS-affiliated HyLo 2002 (http://floc02.diku.dk/HYLO) which was held as part of FLoC 2002, Copenhagen, Denmark. If you are unsure whether your work is of relevance to the workshop, please do not hesitate to contact the workshop organizers for more information. Contact details are given below. For more general background on hybrid logic, and many of the key papers, see the Hybrid Logics homepage (http://hylo.loria.fr/). INVITED SPEAKERS: Patrick Blackburn (INRIA Lorraine, France) Title: Hybrid Logic and Temporal Semantics Valeria de Paiva (PARC, USA) Title: Constructive Hybrid Logics and Contexts Ian Horrocks (University of Manchester, UK) Title: Hybrid Logics and Ontology Languages PROGRAM COMMITTEE: Carlos Areces (INRIA Lorraine, France) Patrick Blackburn (INRIA Lorraine, France) Thomas Bolander (Technical University of Denmark) Torben Bra=FCner (Roskilde University, Denmark) --- Chair Valeria de Paiva (PARC, USA) Melvin Fitting (Lehman College, New York, USA) Balder ten Cate (University of Amsterdam, The Netherlands) J=F8rgen Villadsen (Roskilde University, Denmark) SUBMISSIONS: We invite the contribution of papers reporting new work from researchers interested in hybrid logic. The revised version of accepted papers will be published online in an Elsevier ENTCS (http://www.elsevier.com/locate/entcs/) volume devoted to FLoC 2006 satellite workshops. A preliminary version of the proceedings will also be distributed at the workshop. One author for each accepted paper must attend the workshop in order to present the paper. Please use the HyLo 2006 submission page (http://www.easychair.org/HyLo2006/), handled by the EasyChair conference system, to submit papers. Papers should not exceed 15 pages including references. Authors are strongly encouraged to prepare their submissions according to the ENTCS guidelines (http://www.entcs.org/). IMPORTANT DATES: Deadline for submissions: May 26, 2006 Notification of acceptance: June 23, 2006 Deadline for final versions: July 21, 2006 CONTACT DETAILS: See the workshop homepage (http://hylomol.ruc.dk/HyLo2006) for further information. Please send all correspondence regarding the workshop to the organizers: Patrick Blackburn http://www.loria.fr/~blackbur/ Thomas Bolander http://www.imm.dtu.dk/~tb/ Torben Bra=FCner --- Chair http://www.ruc.dk/~torben/ Valeria de Paiva http://www.cs.bham.ac.uk/~vdp/ J=F8rgen Villadsen http://www.ruc.dk/~jv/ From rrosebru@mta.ca Tue May 16 22:01:15 2006 -0300 From: Thomas Streicher Subject: regular monos not closed under composition To: categories@mta.ca Date: Tue, 16 May 2006 22:20:31 +0200 (CEST) MIME-Version: 1.0 Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII Status: O X-Status: X-Keywords: X-UID: 261 I know that regular monos needn't be closed under composition. One example being the category of monoids. Are there other *natural* examples, e.g. of topological kind? Thomas Streicher From rrosebru@mta.ca Tue May 16 22:02:42 2006 -0300 Mime-Version: 1.0 (Apple Message framework v749.3) To: categories@mta.ca Content-Type: multipart/mixed; boundary=Apple-Mail-4--134679128 From: Rob Goldblatt Subject: reprint Date: Wed, 17 May 2006 11:55:28 +1200 Content-Transfer-Encoding: 7bit Content-Type: text/plain;charset=US-ASCII;delsp=yes;format=flowed Status: O X-Status: X-Keywords: X-UID: 262 The book Topoi: The Categorial Analysis of Logic by Robert Goldblatt has now been reprinted as a relatively inexpensive Dover paperback, available from http://store.yahoo.com/doverpublications/0486450260.html From rrosebru@mta.ca Wed May 17 13:13:37 2006 -0300 Date: Wed, 17 May 2006 10:47:01 +0100 (BST) From: "Prof. Peter Johnstone" To: categories@mta.ca Subject: Re: regular monos not closed under composition MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: O X-Status: X-Keywords: X-UID: 263 On Tue, 16 May 2006, Thomas Streicher wrote: > I know that regular monos needn't be closed under composition. One example > being the category of monoids. Are there other *natural* examples, e.g. of > topological kind? > > Thomas Streicher > Depends what you mean by topological. In the category of locales, regular epis aren't closed under composition (equivalently, regular monos aren't closed under composition in frames): this was proved by Till Plewe, and is in his paper Quotient maps of locales' in Appl. Cat. Struct. 8 (2000), 17--44. Another natural' example (which I regularly use when teaching category theory to graduate students) is regular epis in Cat. Peter Johnstone From rrosebru@mta.ca Thu May 18 10:30:31 2006 -0300 From: "Jonathon Funk" To: Subject: job at Cave Hill Date: Wed, 17 May 2006 16:01:49 -0400 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Status: O X-Status: X-Keywords: X-UID: 264 Job opening at University of West Indies, Cave Hill campus, Barbados. The advertised May 22 deadline is not firm, and may be extended as much = as 1 month. http://www.cavehill.uwi.edu/vacancies/2006/Mar/Professor-Senior%20Lecture= r%20in%20Mathematics.htm -Jonathon Funk From rrosebru@mta.ca Thu May 18 10:33:06 2006 -0300 Mime-Version: 1.0 (Apple Message framework v749.3) Content-Transfer-Encoding: 7bit Content-Type: text/plain; charset=US-ASCII; delsp=yes; format=flowed To: categories@mta.ca From: Rob Goldblatt Subject: Topoi reprint (resent) Date: Thu, 18 May 2006 18:27:34 +1200 Status: O X-Status: X-Keywords: X-UID: 265 The book Topoi: The Categorial Analysis of Logic by Robert Goldblatt has now been reprinted as a Dover paperback, and is currently available for under US$19 from http://store.yahoo.com/doverpublications/0486450260.html or Amazon http://www.amazon.com/gp/product/0486450260/ref=wl_it_dp/ 102-9723571-5488919?% 5Fencoding=UTF8&colid=OXWPP5ID1A26&coliid=I1D9VOS1VZB8DY&v=glance&n=2831 55 or Walmart http://www.walmart.com/catalog/product.do?product_id=4531888 [apologies to those whose received an empty message when this was sent yesterday] From rrosebru@mta.ca Thu May 18 10:34:44 2006 -0300 Date: Thu, 18 May 2006 10:29:55 +0200 From: "Urs Schreiber" To: categories@mta.ca Subject: Hecke eigensheaves and KV 2-vectors MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Status: O X-Status: X-Keywords: X-UID: 266 Dear category theorists, if you don't mind, I'd like to mention a naive observation. Attention of physicists, like myself, has recently been drawn to the geometric Langlands conjecture - since E. Witten and A. Kapustin have pointed out how it can be understood in terms of 2-dimensional topological field theory. Even after having been introduced to some basics, I hardly know anything about geometric Langlands. But I believe I do understand some aspects of 2D topological field theory. In particular, I am fond of the general fact that where \C-modules (\C =3D = the complex numbers) appear in 1D quantum field theory (quantum mechanics), we see C-modules appear in 2D TFT, where now C is some abelian monoidal category. In the most accessible cases of topological field theory we have = C =3D Vect. Now, the 2-category (bicategory) Vect-Mod contains that of Kapranov-Voevodsky 2-vector spaces, but is larger than that (isn't it?). In general, it should make some sense to address objects in C-Mod (module categories for C) as categorified vector spaces; and 1-morphisms in C-Mod a= s categorified linear maps between these. Interestingly, when one studies 2D quantum field theory (topological or conformal), one finds that boundary conditions of the theory (known as "D-branes") are described by objects of objects of C-Mod, i.e. objects of C-module categories. In the above terminology these would be like categorified vectors. Moreover, there are phenomena called "defect lines" or "disorder operators" in 2D QFT. These are known to be described by 1-morphisms in C-Mod, i.e. by categorified linear maps. Therefore a "defect line" may be applied to a "D-brane", much like a linear map may be applied to a vector. The above analogy naturally motivates to contemplate the case where the D-brane is an eigenvector under this action, i.e. where it is sent by the action of the defect line to itself, up to tensoring with an element in C. This might be nothing but a play with words. But, remarkably, Witten and Kapustin point out that the Hecke eigensheaves appearing in the context of geometric Langlands are precisely to be identified with certain D-branes that are categorified eigenvectors of some defect line, in the above sense. Of course, they do not say so using category theoretic terminology. They ar= e addressing an audience of physicists. At one point they apologize for mentioning the term "functor" once. Therefore I was wondering what people knowledgeable in (higher) category theory would think of this. Does my observation make sense? (Of course I am glossing over a couple of technical details.) If yes, has it been observed before? Is it useful for anything? I'd be grateful for any kind of comments. Best regards, Urs Schreiber P.S. As before on previous occasions, I have written up some informal notes with slightly more details on what I have in mind here: http://golem.ph.utexas.edu/string/archives/000810.html From rrosebru@mta.ca Thu May 18 13:59:47 2006 -0300 MIME-Version: 1.0 Subject: Gaunce Lewis Date: Thu, 18 May 2006 09:59:05 -0500 From: "Anthony Elmendorf" To: Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Status: O X-Status: X-Keywords: X-UID: 267 It is with great sadness that I announce that Gaunce Lewis died at home = yesterday after a long struggle with brain cancer. =20 Tony Elmendorf aelmendo@calumet.purdue.edu From rrosebru@mta.ca Fri May 19 08:25:48 2006 -0300 Subject: Re: Hecke eigensheaves and KV 2-vectors To: categories@mta.ca (categories) Date: Thu, 18 May 2006 08:30:57 -0700 (PDT) From: "John Baez" MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Status: O X-Status: X-Keywords: X-UID: 268 Hi - > Now, the 2-category (bicategory) Vect-Mod contains that of > Kapranov-Voevodsky 2-vector spaces, but is larger than that (isn't it?). Yes. Yetter's paper "Categorical Linear Algebra: a Setting for Questions from Physics and Low-Dimensional Topology" characterizes the Kapranov-Voevodsky 2-vector spaces among the Vect-modules. (Here Vect stands for the category of finite-dimensional vector spaces as a monoidal category with its usual tensor product.) In Proposition 25, he says that the 2-category of 2-vector spaces is isomorphic to the 2-category of "finitely semisimple" Vect-modules. These are Vect-modules with a finite set of simple objects such that every object is a finite direct sum of these. We call this a "basis" of simple objects. (I would feel more comfortable if he had said these 2-categories were 2-equivalent, and this is all I really care about.) There are a bunch of obvious Vect-modules that aren't 2-vector spaces in Kapranov and Voevodsky's sense. One is the category VECT of not-necessarily-finite-dimensional vector spaces: not every object is a *finite* direct sum of copies of C. Another is the category Vect^{infinity} for your favorite infinite cardinal "infinity": the same problem, but for a different reason. These counterexamples are a bit dull, in that one imagines there could be some definition of an "infinite-dimensional" 2-vector space that they might fit. But, it seems that that there are two different kinds of infinite-dimensionality at work here: VECT is "tall", while Vect^{infinity} is "wide". There's also VECT^{infinity}, which is tall and wide. So, somebody should straighten this stuff out. Other counterexamples are neither "tall" nor "wide". For example, the category of finite-dimensional representations of a non-semisimple algebra like the algebra of upper triangular matrices. Or, the category of representaions of a quiver. (I think representations of the A_n quiver are the same as representations of the algebra of upper triangular matrices.) I'd be interested to know if any of these sorts of counterexamples is useful in Witten's work, or he can live with 2-vector spaces. > Therefore I was wondering what people knowledgeable in (higher) category > theory would think of this. Does my observation make sense? It makes sense to me! Other mathematicians often respond better to more precisely phrased questions. :-) > If yes, has it been observed before? Not that I know of, but I could be missing examples from other contexts. Certainly category theorists think about "weak fixed points" of endofunctors T: C -> C, that is, objects c in C equipped with isomorphisms Tc -> c. The category of all these weak fixed points is something people enjoy studying: it's an example of a "weak equalizer" and on a good day I know lots of examples. You're asking about something more general: "weak eigenobjects" of endofunctors T: C -> C on *monoidal* categories C. In other words: objects c in C equipped with isomorphisms Tc -> a tensor c for some fixed object a. I bet some people have studied these in some context or other. You may have scared these people away with all your Langlands, branes and defect lines. Category theorists are dual to ordinary people: they often get more confused when you surround an abstract concept with a lot of distacting specifics. "Could you please not give me an example, to help me understand what you're saying?" :-) Best, jb From rrosebru@mta.ca Fri May 19 08:26:55 2006 -0300 Subject: Re: Hecke eigensheaves and KV 2-vectors To: categories@mta.ca (categories) Date: Thu, 18 May 2006 08:37:27 -0700 (PDT) From: "John Baez" MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Status: O X-Status: X-Keywords: X-UID: 269 I wrote: > You're asking about something more general: "weak eigenobjects" of > endofunctors T: C -> C on *monoidal* categories C. In other words: objects > c in C equipped with isomorphisms Tc -> a tensor c for some fixed object > a. Sorry! Actually Urs is asking about an even more general situation. A is a monoidal category, C is a category tensored over A, T: C -> C, and we seek objects c in C equipped with isomorphisms Tc -> a tensor c for some fixed object a. We can form the category of these, which is the pseudo-equalizer of T and a tensor -, and call it an "eigenspace of T". For Urs, A = Vect and C is a Vect-module. Best, jb From rrosebru@mta.ca Sat May 20 10:28:39 2006 -0300 Date: Fri, 19 May 2006 14:39:33 +0200 From: "Urs Schreiber" To: categories Subject: Re: Hecke eigensheaves and KV 2-vectors MIME-Version: 1.0 Content-Type: text/plain; charset=ISO-8859-1; format=flowed Content-Transfer-Encoding: quoted-printable Content-Disposition: inline Status: O X-Status: X-Keywords: X-UID: 270 On 5/18/06, John Baez wrote: > I'd be interested to know if any of these sorts of counterexamples is > useful in Witten's work, or he can live with 2-vector spaces. > I think what is needed in Witten's context is the following. Fix some topological space X. In the Witten/Langlands examples this is the moduli space X =3D Bun_G of G-bundles over some complex curve. Hm, probably it has more structure than just being topological. But anyway. The relevant object of Vect-Mod (the relevant 2-vector space) is the category of vector bundles over X. Of course this is a slight lie. Really we want to be working with locally free (coherent?) sheaves on X. But anyway. So a 2-vector here is a vector bundle over X. For Kaparanov-Voevodksy we would instead take X to be a finite set. Next, the endomorphisms that are relevant are spans over X, i.e spaces with two (possibly different) projections onto X. Certainly these "spaces" shoul= d carry some appropriate extra structure. For the application to Langlands these spans are usually called Hecke_x . We act with these spans on our 2-vectors (vector bundles) by pulling back along one leg and pushing forward along the other. For the Kaparanov-Voevodksy case our spans would be vector bundles with two projections to some finite set,. i.e matrices whose entries are vector spaces. > Therefore I was wondering what people knowledgeable in (higher) category > > theory would think of this. Does my observation make sense? > > It makes sense to me! Other mathematicians often respond better to > more precisely phrased questions. :-) I am aware that the question is a little unspecific. The best thing about this observation is that it fits so very nicely into the general picture that is emerging in 2D QFT. Actually Urs is asking about an even more general situation. > A is a monoidal category, C is a category tensored over A, > T: C -> C, and we seek objects c in C equipped with isomorphisms > Tc -> a tensor c for some fixed object a. And, moreover, T is "A-linear". That's kind of important. We can form the category of these, which is the pseudo-equalizer > of T and a tensor -, and call it an "eigenspace of T". And that's where Hecke eigensheaves, and hence the corresponding "D-branes"= , should live in. I think. From rrosebru@mta.ca Sat May 20 10:29:16 2006 -0300 To: categories@mta.ca Subject: jobs: 1 PhD and 1 Post-doc position at CWI, Amsterdam Date: Fri, 19 May 2006 18:33:38 +0200 (CEST) From: F.S.de.Boer@cwi.nl (Frank de Boer) Status: O X-Status: X-Keywords: X-UID: 271 ======================================================================= CWI: 1 PhD Position and 1 Postdoc Position in the CREDO Project ======================================================================= The theme SEN3 Coordination Languages at CWI (http://www.cwi.nl) has 1 PhD position for four years and 1 postdoc position for three years. Both positions are within the IST-33826 research CREDO project "Modeling and analysis of evolutionary structures for distributed services", recently funded by the European Union. This project starts September 1, 2006, and lasts for three years. The objective of this project is the development and application of an integrated suite of tools for compositional modelling, testing, and validation of software for evolving networks of dynamically reconfigurable components. More information on this project can be found at the following webpage: http://www.cwi.nl/projects/credo. Job description --------------- The candidates are expected to work on the design and development of a new high-level modelling language Credo for the dynamic composition of highly reconfigurable component-based software systems, and light-weight and automated verification techniques and tools supporting this language. The main overall idea of Credo is the integration of a formal component model based on concurrent objects which support run-time updates with a new model of component connectors based on mobile channels. The main objective of this integration is to support rapid prototyping and automated validation of networks of distributed services implemented by components, focusing on analyzing the effect of dynamic reconfiguration. The kernel of the Credo tool will consists of an abstract interpreter for the language implemented in the rewriting logic of the Maude system (http://maude.cs.uiuc.edu). The abstract interpreter forms the basis for the development and integration of simulation, testing and validation tools for the compositional analysis of functional, timing, and resource requirements. Bothe candidates should have a background in software engineering, concurrency and distributed systems, and practical software development or formal methods. The PhD candidate should have at least a master degree in computer science and the candidate for the postdoc position should have a PhD degree in computer science. Scientific Cluster of Software Engineering at CWI ------------------------------------------------- CWI is an internationally renowned research institute in mathematics and computer science, located in Amsterdam, The Netherlands. The theme SEN3 Coordination Languages of the Scientific Cluster of Software Engineering at CWI is a dynamic group of internationally recognized researchers who work on Coordination Models and Languages and Component-Based Software Composition. The activity in SEN3 is a productive, healthy mix of theoretical, foundational, and experimental work in Computer Science, ranging in a spectrum covering mathematical foundations of models of computation, formal methods and semantics, implementation of advanced research software systems, as well as their real-life applications. Terms of employment ------------------- The salary and terms at CWI are in accordance with the Dutch "CAO-onderzoeksinstellingen". The initial salary for a Ph.D. student is Euro 1848,- gross per month. The salary grows to Euro 2367,- in the fourth year. The salary is supplemented with a holiday bonus of 8% and an end-of-year bonus of 4.1 %. The initial salary for a postdoc is around Euro 2803,- (Euro 2848,- per 1/8/06) gross per month. The salary is supplemented with a holiday bonus of 8% and an end-of-year bonus of 4.1 %. CWI offers very attractive working conditions, including flexibility and help with housing for foreigners. Further details --------------- More information can be obtained Dr. F.S. D Boer, telephone +31-20-592-4139, email F.S.de.Boer 'at' cwi.nl. How to apply ------------ To apply, please send a statement of your interest, together with curriculum vitae, letters of references, and possibly list of publications to F.S.de.Boer 'at' cwi.nl. Make sure that you specifically mention the CREDO project. From rrosebru@mta.ca Mon May 29 16:00:25 2006 -0300 Date: Mon, 29 May 2006 06:20:02 -0400 (EDT) From: Peter Freyd Message-Id: <200605291020.k4TAK2pr007529@saul.cis.upenn.edu> To: categories@mta.ca Subject: Bill Hatcher -- an obituary Status: O X-Status: X-Keywords: X-UID: 272 [Bill's category papers were concerned mostly with categories of algebras, his non-mathematical works with the spreading of the Baha'i faith.] The Globe and Mail 29 May 2006 LIVES LIVED Facts & Arguments William S. Hatcher Roshan Danesh 663 words A18 Husband, father, mathematician, philosopher, pursuer of authenticity. Born Sept. 20, 1935, in Charlotte, N.C. Died Nov. 27, 2005, in Stratford, Ont., of a heart attack, aged 70. I always looked up to my uncle, William Hatcher. Truly, almost everyone who met him looked up to him. At 6-foot-6, and with the physical size and booming voice that betrayed his roots in the deep south of the United States, he was pretty much always the largest presence wherever he found himself. But for my uncle Bill, physical presence was merely the reflection of the other ways in which he was a towering figure. His mind was what made him most well-known. He brought an iron-clad precision and clarity to the most ubiquitous, essential, and seemingly eternal questions of human life. In his early 20s, he began contemplating a logical proof of the existence of God. When he fully completed the proof some decades later, he travelled over many years to universities across Canada, the United States, and Europe, where he explicated the proof to packed audiences. The logical proof was one of Bill's most characteristic achievements. It was the perfect blending of the two passions of his mind science and religion. A mathematician and philosopher at Universiti Laval for more than 30 years, he was listed as one of the eight Platonist philosophers of the latter half of the 20th century in the respected Encylopidie Philosophique Universelle. His contribution to the study of religion, and in particular the Baha'i faith a subject on which he co-authored the seminal introductory text led him to be recognized as one of the greatest scholars of Baha'i studies. But it was uncle Bill's qualities of heart that I will remember most. He was staunchly committed to the pursuit of authentic relationships and the principle that love is expressed through self-sacrifice, service, altruism, and putting others ahead of oneself. He captured this theme in his book Love, Power, and Justice: The Dynamics of Authentic Morality. One time he said I should simply look around me when walking down the street and reflect on my reaction to those I saw. He was giving me a lesson about how to stay conscious of my own prejudices, my own self-interested commitments which, he insisted, were reflections of a failure to authentically love and relate to other human beings. Uncle Bill constantly pursued authenticity in his own life, and strived to live a life characterized by service to others. Of course, he sometimes failed, but he often succeeded. Sometimes his actions took the form of grand gestures such as moving to Russia at a dangerous time because there were contributions he could make as the society emerged from the communist era. Other times it was in the simple sweetness of how he always was prepared to listen to one's ideas, share his thoughts, and encourage one to do better. Like all high-achievers, Bill was constantly striving to do better. In the last few years, he seemed to become a little doubtful of whether his myriad accomplishments (raising three children with his wife Judith, training thousands of young minds, publishing more than 50 books and articles, being a leader of the Baha'i community of Canada and one of its leading scholars) were enough. To those of us who knew him intimately and loved him dearly, it was hard to take such insecurities seriously. My uncle Bill touched the hearts of thousands of people, and helped educate, refine, and inspire their minds. The gifts he gave others through his writings and teaching, and his acts of kindness, were more than we could fairly expect from anybody. The gift he gave me was clear: Be humble, strive as hard as you can, and be a lover of humanity. I will cherish that gift forever. Roshan is William Hatcher's nephew. Illustration From rrosebru@mta.ca Mon May 29 16:01:20 2006 -0300 Date: Mon, 29 May 2006 08:50:12 -0500 From: Joseph Goguen Festschrift To: categories@mta.ca Subject: Joseph Goguen Festschrift -- Call for Participation Content-Type: text/plain; charset=us-ascii Content-Disposition: inline Status: O X-Status: X-Keywords: X-UID: 273 Algebra, Meaning, and Computation: A Festschrift Symposium in Honor of Joseph Goguen THIRD CALL FOR PARTICIPATION !!!URGENT: HOTEL ROOMS MUST BE BOOKED SOON!!! Joseph Goguen is one of the most prominent computer scientists worldwide. His numerous research contributions span many topics and have changed the way we think about many concepts. Our views about data types, programming languages, software specification and verification, computational behavior, logics in computer science, semiotics, interface design, multimedia, and consciousness, to mention just some of the areas, have all been enriched in fundamental ways by his ideas. A Symposium to honor Joseph Goguen on his 65th birthday on June 28, 2006 will be held at the Computer Science and Engineering Department of the University of California in San Diego (UCSD) in June 27-29 2006. There will also be an IFIP WG1.3 meeting co-located with the Symposium that will take place on June 30. Leading researchers in the different areas spanned by Joseph Goguen's work have contributed over thirty papers to the Festschrift volume being published by Springer-Verlag. These papers will be presented at the Symposium in a three full day programme. A Symposium banquet is planned for June 28 to celebrate Joseph Goguen's 65th Birthday. The Symposium organizers welcome participation by any members of the scientific community who would like to take part in this important event. LOCAL INFORMATION AND VENUE The UCSD Campus is located in La Jolla, on top of a spectacular mesa overlooking the Pacific Ocean. The Symposium, and also the IFIP WG1.3 meeting, will be held at UCSD's Computer Science Department. The town of La Jolla is one of the most beautiful in the California Coast. Weather in June should be very pleasant, typically around 25 degrees Celsius, mostly sunny but possibly with some morning coastal fog. La Jolla is easy to reach from the San Diego Airport (see below). All major US airlines fly into San Diego from other main hubs like Los Angeles, San Francisco, Washington DC, New York, Chicago, Dallas, and so on. The Festschrift Symposium in Honor of Joseph Goguen is being hosted by the Jacobs School of Engineering's Department of Computer Science and Engineering. The meetings will be held in the Computer Science and Engineering Building's main Auditorium, room 1202. For a map of the campus and directions to UCSD CSE Department see http://formal.cs.uiuc.edu/goguenFest/ SYMPOSIUM HOTEL A block of rooms at a special reduced rate been reserved for this event at the Best Western La Jolla Inn by the Sea, which is conveniently located at the heart of downtown La Jolla and very close to the Ocean. -Website: http://lajollainnbythesea.com/ -Total is $139.00/night (approx.$153.59/night after tax) -Reservation Deadline: May 26th: all unbooked rooms will be released on May 26th -Parking fees at the hotel have been waived for our group How to make a reservation. Call: 800-462-9732 (from within the U.S.) 800-526-4545 (from within California and Canada) Email: innbyseabw@aol.com Mention confirmation #203732 or UCSD/Computer Science & Engineering (Note: a previous mistake about the hotel phones has been corrected) Transportation from the San Diego airport to the Best Western Hotel (Both terminals 1 & 2) is provided by Cloud 9 Shuttle for $16.00. This is the cheapest way to get to the hotel and the fee can be directly charged to the hotel room. The shuttle runs 24 hours a day. Taxi or rental car are alternative options. SYMPOSIUM REGISTRATION The early registration fee of$255.00 - gives access to the entire Festschrift event. This includes meetings on June 27 - 29, snacks and drinks during the meeting, lunches on all 3 days, Festschrift volume, Concert with Wine & Cheese Reception on June 27th, banquet on June 28th and transportation to and from hotel for events. The registration fee of $51.00 for the IFIP WG 1.3 Meeting on June 30th includes snacks & drinks during the meeting, lunch, and transportation to and from hotel. The registration can be made using a credit card at the symposium registration website: https://secure.ucsd.edu/external/cse-festschrift/ Early registration is strongly encouraged. A registration fee of$300 will be required after June 11, and a late registration fee of \$350 after June 21st. BOOK SOON! Participants are strongly encouraged to register as soon as possible, and also to proceed immediately with booking of their hotel room and airline tickets. June is a high tourist season in La Jolla, and late booking of airline tickets, or of hotel rooms not under the special conditions mentioned above, can be difficult and significantly more expensive. SYMPOSIUM PROGRAM A preliminary version of the Symposium schedule can be found in http://formal.cs.uiuc.edu/goguenFest/ ORGANIZATION Program Committee Kokichi Futatsugi, JAIST, Japan Jean-Pierre Jouannaud, Ecole Polytechnique, France Jose Meseguer, University of Illinois at Urbana-Champaign, USA Local Chair Keith Marzullo, UCSD, USA Main Symposium Coordinator Ms. Briana Ronhaar Computer Science & Engineering University of California, San Diego 9500 Gilman Dr. #0404 La Jolla, CA 92093-0404 Email: bronhaar@cs.ucsd.edu Phone: ++1-858-822-5198 Fax: ++1-858-822-1559 From rrosebru@mta.ca Mon May 29 16:02:01 2006 -0300 Subject: Notes from Glasgow PSSL From: Tom Leinster To: categories@mta.ca Content-Type: text/plain; charset=UTF-8 Date: Mon, 29 May 2006 16:56:00 +0100 Mime-Version: 1.0 Content-Transfer-Encoding: quoted-printable Status: O X-Status: X-Keywords: X-UID: 274 Dear all, Notes from some of the talks at the recent PSSL, including Jon Woolf's three-part introduction to derived categories, are available: http://www.maths.gla.ac.uk/~tl/pssl/#intellectual Contents: Miles Gould, "Coherence for categorified algebraic theories" Panagis Karazeris, "Flatness of functors into sites" J=C3=BCrgen Koslowski, "What is the correct notion of morphism for interpolative semigroups?" Tom Leinster, "A universal Banach space" Jon Woolf, "Derived categories 1, 2, 3" Best wishes, Tom --=20 Tom Leinster From rrosebru@mta.ca Mon May 29 16:02:25 2006 -0300 Date: Mon, 29 May 2006 18:21:59 +0200 From: Carlos Areces MIME-Version: 1.0 To: Carlos Areces Subject: HyLo 2006: Extended Deadline Content-Type: text/plain; charset=3DISO-8859-1; format=3Dflowed Content-Transfer-Encoding: 8bit Status: O X-Status: X-Keywords: X-UID: 275 ******************************************************************* HYLO 2006 DEADLINE EXTENDED! New deadline is Thursday June 1 ******************************************************************* FINAL CALL FOR PAPERS International Workshop on Hybrid Logic 2006 (HyLo 2006) Affiliated with LICS 2006 August 11, 2006, Seattle, USA ******************************************************************* AIMS AND SCOPE: Hybrid logic is a branch of modal logic in which it is possible to directly refer to worlds/times/states or whatever the elements of the (Kripke) model are meant to represent. Although they date back to the late 1960s, and have been sporadically investigated ever since, it is only in the 1990s that work on them really got into its stride. It is easy to justify interest in hybrid logic on applied grounds, because of the usefulness of the additional expressive power. For example, when reasoning about time one often wants to build up a series of assertions about what happens at a particular instant,and standard modal formalisms do not allow this. What is less obvious is that the route hybrid logic takes to overcome this problem (the basic mechanism being to add nominals --- atomic symbols true at a unique point --- together with extra modalities to exploit them) often actually improves the behavior of the underlying modal formalism. For example, it becomes far simpler to formulate modal tableau, resolution, and natural deduction in hybrid logic, and completeness and interpolation results can be proved of a generality that is not available in orthodox modal logic. Hybrid logic is now a mature field, therefore a theme of special interest at this HyLo workshop will be the combination of hybrid logic with other logics, the basic methodological question being "what is the best way of hybridizing a given logic?" However, submissions in all areas of hybrid logic are welcome. The workshop HyLo 2006 is likely to be relevant to a wide range of people, including those interested in description logic, feature logic, applied modal logics, temporal logic, and labelled deduction. The workshop continues a series of previous workshops on hybrid logic, for example the LICS-affiliated HyLo 2002 (http://floc02.diku.dk/HYLO) which was held as part of FLoC 2002, Copenhagen, Denmark. If you are unsure whether your work is of relevance to the workshop, please do not hesitate to contact the workshop organizers for more information. Contact details are given below. For more general background on hybrid logic, and many of the key papers, see the Hybrid Logics homepage (http://hylo.loria.fr/). INVITED SPEAKERS: Patrick Blackburn (INRIA Lorraine, France) Title: Hybrid Logic and Temporal Semantics Valeria de Paiva (PARC, USA) Title: Constructive Hybrid Logics and Contexts Ian Horrocks (University of Manchester, UK) Title: Hybrid Logics and Ontology Languages PROGRAM COMMITTEE: Carlos Areces (INRIA Lorraine, France) Patrick Blackburn (INRIA Lorraine, France) Thomas Bolander (Technical University of Denmark) Torben Bra=FCner (Roskilde University, Denmark) --- Chair Valeria de Paiva (PARC, USA) Melvin Fitting (Lehman College, New York, USA) Balder ten Cate (University of Amsterdam, The Netherlands) J=F8rgen Villadsen (Roskilde University, Denmark) SUBMISSIONS: We invite the contribution of papers reporting new work from researchers interested in hybrid logic. The revised version of accepted papers will be published online in an Elsevier ENTCS (http://www.elsevier.com/locate/entcs/) volume devoted to FLoC 2006 satellite workshops. A preliminary version of the proceedings will also be distributed at the workshop. One author for each accepted paper must attend the workshop in order to present the paper. Please use the HyLo 2006 submission page (http://www.easychair.org/HyLo2006/), handled by the EasyChair conference system, to submit papers. Papers should not exceed 15 pages including references. Authors are strongly encouraged to prepare their submissions according to the ENTCS guidelines (http://www.entcs.org/). IMPORTANT DATES: Deadline for submissions: June 1, 2006 (strict) Notification of acceptance: June 20, 2006 Deadline for final versions: July 21, 2006 CONTACT DETAILS: See the workshop homepage (http://hylomol.ruc.dk/HyLo2006) for further information. Please send all correspondence regarding the workshop to the organizers: Patrick Blackburn http://www.loria.fr/~blackbur/ Thomas Bolander http://www.imm.dtu.dk/~tb/ Torben Bra=FCner --- Chair http://www.ruc.dk/~torben/ Valeria de Paiva http://www.cs.bham.ac.uk/~vdp/ J=F8rgen Villadsen http://www.ruc.dk/~jv/ From rrosebru@mta.ca Mon May 29 16:02:58 2006 -0300 Subject: Higher-dimensional algebra - a language for quantum spacetime To: categories@mta.ca (categories) Date: Mon, 29 May 2006 11:33:57 -0700 (PDT) From: "John Baez" MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Status: O X-Status: X-Keywords: X-UID: 276 Dear Categorists: Some of you might like to see the transparencies of this talk: http://math.ucr.edu/home/baez/quantum_spacetime/ It's a general colloquium talk, where I'll try to explain to the physicists at the Perimeter Institute how higher categories show up in physics. You won't learn any category theory here. If you want details, there are links to some papers. Best, jb ............................................................................. Higher-dimensional algebra: a language for quantum spacetime Category theory is a general language for describing things and processes - called "objects" and "morphisms". In this language, the counterintuitive features of quantum theory turn out to be properties that the category of Hilbert spaces shares with the category of cobordisms - in which objects are choices of "space", and morphisms are choices of "spacetime". The striking similarities between these categories suggests that "n-categories with duals" are a promising framework for a quantum theory of spacetime. We sketch the historical development of these ideas from Feynman diagrams, to string theory, topological quantum field theory, spin networks and spin foams, and especially recent work on open-closed string theory, quantum gravity coupled to point particles, and 4d BF theory coupled to strings. From rrosebru@mta.ca Mon May 29 20:47:44 2006 -0300 To: categories@mta.ca Subject: two topology questions From: Paul Taylor Date: Mon, 29 May 2006 21:23:18 +0100 Status: O X-Status: X-Keywords: X-UID: 277 Is there an introduction to CONTINUOUS LATTICES written specifically for (and ideally BY) real analysts? (completely unrelated:) Is there a word for a (necessarily compact) space X for which the unique map X --> 1 is a proper surjection? Maybe "occupied"? In the latter question, you may interpret "space" in any appropriate sense, for example to mean locale or topos or ASD object or formal space, or whatever you like. Paul PS See you all in Canada (Calgary & Nova Scotia) in June. From rrosebru@mta.ca Wed May 31 22:06:26 2006 -0300 From: "Ronnie Brown" To: Subject: some Bangor preprints Date: Wed, 31 May 2006 22:41:05 +0100 Content-Type: text/plain;charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable Status: O X-Status: X-Keywords: X-UID: 278 I have put some preprints on http://www.bangor.ac.uk/~mas010/brownpr.html 1. Normalisation for the fundamental crossed complex of a simplicial set. R. Brown and R. Sivera=20 This is but one more step in setting up crossed complexes as a basic somewhat nonabelian tool in algebraic topology. It shows how crossed complexes are used to express the homotopy addition lemma, and gives an analogue of a theorem classical for simplicial abelian groups. 2. Analogy, concepts and methodology in Mathematics. R. Brown and T. Porter This is an an article for undergraduates to appear in Eureka, Journal of the Archimideans at Cambridge. It is about rather than on mathematics. It includes two quotations from a correspondence with Grothendieck. It could be seen as relevant to the article `Mathematical Morality' by Eugenia Cheng. Other Bangor preprints are on http://www.informatics.bangor.ac.uk/public/mathematics/research/preprints/ Ronnie Brown www.bangor.ac.uk/r.brown/topgpds.html