Seminars of the CENTRE de RECHERCHE en THEORIE des CATEGORIES CATEGORY THEORY RESEARCH CENTER C ---------> R | / | | / | | / | | / | v / v T ---------> C Tuesday 18 January 1994 2:30 - 4:00 pm J. Lambek (McGill) "Quaternions and Physics" Tuesday 25 January 1994 2:30 - 4:00 pm M. Barr (McGill) "Weak Acyclic Models" ABSTRACT: Weak Acyclic Models There have been two categorical versions of acyclic models. The first has assumed the existence of natural transformations as input and produced natural transformations, unique up to homotopy as output. The second begins with an isomorphism in 0 homology and produces a homology isomorphism in all degrees, but does not extend maps other than isomorphisms, nor does it produce isomorphisms that are either unique or natural. Moreover, although both versions are called acyclic models, their proofs were entirely unrelated. Recently, I have discovered a new version (and new proof) of acyclic models of which each of the previous versions were special cases, but that always produces unique-up-to-homotopy maps (in a category of fractions, in general) and these maps are always natural. Moreover, there is at least one other specialization that falls between the other two in strength and is interesting in its own right. I give an elegant proof of the equivalence of de Rham and singular homology on C^\infty manifolds as an application. Tuesday 1 February 1994 2:30 - 4:00 R. Pare (Dalhousie, visiting UQAM) "Universal Covering Categories" Tuesday 15 February 1994 1:30 - 3:00 RFC Walters (University of Sydney) Distributive automata and asynchronous circuits ABSTRACT: > An important part of concurrency, the study of systems with > many activities occurring together, is the study of > asynchronous circuits. The aim is the analysis and design > of circuits with predictable behaviour from components > which have unspecified delays. > > In this lecture we report the work of a group in Sydney > (Walters, Katis, Khalil, Weld) and Milan (Sabadini, Vigna) > on a model of asynchronous circuits using a structured > kind of automata called distributive automata -- automata > constructed from data-types and their operations using the > operations of a distributive category. > > We give a simple mathematical definition of asynchronous > circuit as a particular type of distributive automaton. The > definition applies not just to Boolean data, and includes > both synchronous circuits and transition logic. As an example > we discuss Sutherland's micropiplines. > > If the data type is the natural numbers all recursive functions > can be computed by delay independent circuits. In fact to every > distributive automaton there is an asynchronous circuit whose > visible behaviour is that of the distributive automaton - this > gives a way of translating programs into delay insensitive circuits. > Tuesday 1 March 1994 2:30 - 4:00 P. Panangaden (McGill) "An Analogy between Feynman Diagrams and Proof Nets" (joint with R. Blute). ABSTRACT: Proof Nets and Feynman Diagrams Linear logic appears to have many curious, possibly co-incidental, connections with quantum field theory. In this talk I will show how one can give an interpretation of proof nets as Feynman diagrams. More porecisely, with each proof nets is associated a formal "integral" constructed inductively from the net. Certain formal rules are introduced for computing with these integrals. We show that the process of simplification of these integrals automatically performs cut elimination. Furthermore, the exponential types are modelled as exponentials. All the equations of Linear Realizability Algebra are validated. The correspondence between nets and the formal integrals is very similar to the correspondence between Feynman diagrams and the integrals corresponding to terms in the perturbation expansion of a Quantum Field Theory. Thus our analogy is not a mere pictorial resemblence of graphical structures but actually extends to a correspondence between integrals. This work is part of an an investigation into the Geometry of Interaction program being carried out by Rick Blute, Phil Scott, Robert Seely and me. The specific results in this talk were obtained by Rick and me last july. Whether these results mean anything significant remains to be seen. Tuesday 8 March 1994 2:30 - 4:00 M. Bunge (McGill) "Lex Completions and the Symmetric Topos" (joint with Aurelio Carboni) ABSTRACT. F.W.Lawvere(1983) discussed the notion of distributions on a Grothendieck topos and asked the question of the existence of a topos (the "symetric topos") whose points are the distributions on a given topos. Using classifying toposes I answered this question affirmatively in "Cosheaves and Distributions on Toposes" (to appear in the Alana Day Issue of Algebra Universalis), for a wider class of categories, namely, the small generated small presented Set-modules. In the present work (joint with Aurelio Carboni and part of a wider project) we concentrate on giving an algebraic (rather than logical) construction of the symmetric topos. Just as in Algebra, the construction of the free commutative monoid on a set - here replaced by the lex completion of a small category - plays a key role. We also prove, using the symmetric topos, a characterization of toposes in the setting of Set-modules. The analogous characterization for frames in the setting of suplattices is also valid but it differs from that of Joyal & Tierney(1984). Also in this case, applications to descent can be derived and in a rather simple manner, using descent theory for modules. Further applications - which stem from the tripleability of the biadjoint pair given by Sigma and the forgetful - include a theorem of Moerdijk(1988) and of Makkai-Pare(1989) on the existence and nature of indexed colimits of toposes. Finally, using unpublished work of A.M.Pitts(1986) on lex completions of small catego categories, we describe lex completions of sites and use this in order to give a site presentation of the symmetric topos of a topos, knowing a site presentation for the latter. Some open questions in this connection will be mentioned. Tuesday 15 March 1994 2:30 - 4:00 J. Lambek (McGill) ``Diagram chasing in Goursat varieties''. Tuesday 22 March 1994 2:30 - 4:00 M. Makkai (McGill) "Catgory theory in and over the realizability topos" Tuesday 29 March 1994 2:30 - 4:00 J. Beck "Elements of simplicial objects" Tuesday 5 April 1994 2:30 - 4:00 J. Lambek (McGill) "A South-Pacific rewrite system" ABSTRACT: (joint with Mira Bhargava) The talk analyzes the rewriting grammar discovered by Lounsbury to account for the data on Trobriand kinship terminology collected by Malinowski. Tuesday 19 April 1994 2:30 - 4:00 J.J.C. Vermeulen (University of Cape Town). "Stably closed maps of locales" ABSTRACT. We inverstigate the properties of stably closed, or proper maps of locales, in a setting formally similar to that developped by A. Joyal and M. Tierney for treating the descent theory of localic open maps. We show that proper maps are precisely the compact (perfect) maps previously considered by P.T. Johnstone, and that proper surjections are stable coequalizers, effective for descent in the category of locales. Tuesday 26 April 1994 2:30 - 4:00 Re'ne' Lavendhomme (Louvain-la-Neuve) "Une notion ge'ne'rale de module croise'" Tuesday 3 May 1994 2:30 - 4:00 J. Otto (McGill) "the linear time hierarchy" Tuesday 10 May 1994 2:30 - 4:00 Giovanni Landis (Trieste) "Noncommutative geometry and finite quantum physics" Tuesday 17 May 1994 2:30 - 4:00 F. Lamarche (Imperial College) "The free monoidal closed category with products." Abstract: Here we mean just what the title says: we describe the free such category, with unit to tensor and terminal object, on a set of objects. It is an application of our theory of proof nets for intuitionistic linear logic, and a good way of motivating and illustrating it. Tuesday 24 May 1994 2:30 - 4:00 M. Barr (McGill) "Oriented singular homology." Tuesday 31 May 1994 2:30 - 4:00 I. Moerdijk (University of Utrecht) "Smooth etendues and holonomy" Abstract: In this lecture we will discuss Grothendieck's notion of an etendue topos, and explain its relation to the holonomy of foliations. Tuesday 7 June 1994 2:30 - 4:00 Hongde Hu (York) ``Quasi-coproducts and accessible categories with wide pullbacks'' (joint work with W. Tholen) Abstract: Most of this lecture will be about connections between quasi-coproducts and wide pullbacks. In particular, we will discuss a duality involving the 2-category of accessible categories with wide pullbacks and accessible functors preserving wide pullbacks. We will also show that a wide pullback-preserving functor of accessible categories with wide pullbacks is accessible iff it satisfies the solution set condition. Tuesday 12 July 1994 2:30 - 4:00 Cristina Pedicchio (Trieste) Maltsev categories and commutator theory *********************************************************** Tuesday 13 September 1994 2:30 - 4:00 M. Barr (McGill) "Top^op is a quasi-variety" Tuesday 27 September 1994 2:30 - 4:00 R.J. Wood (Dalhousie) "The Cantor-Gleason-Dilworth Theorem". Thursday 29 September 1994 2:30 - 4:00 Ayelet Lindenstrauss "Algebraic K-Theory and large-scale geometry". Tuesday 4 October 1994 2:30 - 4:00 A. Ursini (Siena) ``Relational semantics for linear logics" Tuesday 18 October 1994 2:30 - 4:00 pm J. Lambek (McGill) "Relations in Categories" Tuesday 25 October 1994 2:30 - 4:00 pm S. Finkelstein (McGill) "Introduction to logic programming and \tau-categories" (This talk will present introductory material needed in the next week's talk.) Friday 28 October 1994 2:30 - 4:00 pm Ed Keenan (UCLA) "Two results on type theory for natural languages" Tuesday 1 November 1994 2:30 - 4:00 pm S. Finkelstein (McGill) "Logic programming in \tau-categories" Tuesday 8 November 1994 2:30 - 4:00 pm Prakash Panangaden (McGill) "A logical view of concurrent constraint programming" ABSTRACT: The Concurrent Constraint Programming paradigm has been the subject of growing interest as the focus of a new paradigm for concurrent computation. Like logic programming it claims close relations to logic. In fact these languages {\em are} logics in a certain sense that we make precise in this talk. In recent work it was shown that the denotational semantics of determinate concurrent constraint programming languages forms a hyperdoctrine, and so the combinators of determinate concurrent constraint programming can be viewed as logical connectives. In the present talk we extend these ideas to the operational semantics of these languages and thus make available similar analogies for a much broader variety of languages including the indeterminate concurrent constraint programming languages and concurrent block-structured imperative languages. Tuesday 29 November 1994 2:30 - 4:00 pm D Cubric (Ottawa) "The Universal Fragment of Intuitionistic Logic" Tuesday 6 December 1994 2:30 - 4:00 pm R Blute (Ottawa) "Linear Lauchli Semantics." PLACE: BURNSIDE HALL 920, McGILL UNIVERSITY (COOKIES AND COFFEE AS USUAL AFTER THE TALK).)