Seminars of the CENTRE de RECHERCHE en THEORIE des CATEGORIES CATEGORY THEORY RESEARCH CENTER C ---------> R | / | | / | | / | | / | v / v T ---------> C Tuesday, 22 January 2002 2:30 - 4:00 Eduardo Ochs A system of natural deduction for categories Abstract: We will present a logic (system NDC) whose terms represent categories, objects, morphisms, functors, natural transformations, sets, and points, and whose deductions represent certain constructive operations involving those entities; certain categorical facts are ``essentially syntactical'', in the sense that such a fact, A, can be proved by expressing it as a term of system NDC, A', then looking for a deduction for A' in NDC, then filling up the missing details. Deductions in NDC are quite short, but NDC ``sees only the syntactical side and ignores the equational side'' --- for example, the rule that constructs a functor in NDC doesn't check if the resulting entity respects identities and compositions, so it builds only a ``proto-functor''; the equational conditions must be checked in separate. The presentation will be focused on formalizing system NDC, and on analyzing a few toy examples --- mainly the syntactical proofs of the Yoneda Lemma, and of all the steps of the demonstration that a category of the form Set^C is an elementary topos. Tuesday, 29 January 2002 2:30 - 4:00 Colin McLarty (Department of Philosophy, Case Western Reserve University) Logic and geometry from Eilenberg-MacLane to Grothendieck Tuesday, 5 February 2002 2:30 - 4:00 Marta Bunge Extensive 2-categories and Top Tuesday, 12 February 2002 2:30 - 4:00 Marta Bunge A Van Kampen theorem for toposes and applications Tuesday, 19 February 2002 2:30 - 4:00 Marta Bunge Locally constant objects in a Grothendieck topos Tuesday, 5 March 2002 2:30 - 4:00 Mihaly Makkai On syntax: an exhortation Abstract: The recent discussion on Syntax on the Categories list exhibited a surprising disregard for the achievements of Mathematical Logic in the subject of Syntax, and thereby for the only existing system of thought that can be called a science of Syntax. (This statement is subject to possible errors insofar I may have missed some relevant contributions to the discussion.) This disregard is particularly relevant in the context of Categorical Logic. My papers "Generalized sketches as a framework for completeness theorems", JPAA 115 (1997), pp. 49-79, 179-221 and 241-274, are intended as a principled approach to the syntax of Categorical Logic in general, one that does take Mathematical Logic into account (I am unaware of other relevant work that would address the same issues). In the talk, I will discuss, among others, why the concern with completeness theorems is natural. As a particular instance, the General Completeness Theorem of the work is used to give a uniform specification of the syntax of well-defined programs in any one of the programming languages based on categorical doctrines such as (1) the product/coproduct doctrine (D-sketches), (2) the finite-limit doctrine, (3) the Cartesian-closed doctrine, and many possible others. Other known specifications in these areas are of interest, but they are unsatisfactory insofar they do not remain in the original categorical framework, but translate it into a traditionalist linguistic framework in an ad hoc, case-by-case, manner. Tuesday, 12 March 2002 2:30 - 4:00 Mihaly Makkai On syntax: examples (a "workshop") ABSTRACT: (and on the seminar web page) Tuesday, 26 March 2002 2:30 - 4:00 Susan Niefield (Union College) Homotopy pullbacks, Lax pullbacks, and Exponentials Tuesday, 2 April 2002 2:30 - 4:00 Richard Squire The Definability of transition surjective presheaves Part I: The axiom of constant domain Tuesday, 9 April 2002 2:30 - 4:00 Richard Squire The Definability of transition surjective presheaves Part II: The reduction to subobjects of Omega Tuesday, 16 April 2002 2:30 - 4:00 Jean-Pierre Marquis From a geometrical point of view: the categorical perspecive on mathematics and its foundations Tuesday, 23 April 2002 2:30 - 4:00 M Barr Absolute homology Abstract Call two maps, f,g: C --> C', of chain complexes absolutely homologous if for any additive functor F, the induced Ff and Fg are homologous (induce the same map on homology). It is known that the identity is absolutely homologous to 0 iff it is homotopic to 0 and tempting to conjecture that f and g are absolutely homologous iff they are homotopic (the "if" part is obvious). This conjecture is false, but there is an equational characterization of absolute homology. I also characterize left absolute and right absolute (in which F is quantified over left or right exact functors). Tuesday, 30 April 2002 2:30 - 4:00 2:30 - 4:00 Tsemo Aristide Stacks and affine manifolds. Tuesday, 7 May 2002 2:30 - 4:00 Alexander Nenashev Simplicial constructions in the K-theory of exact categories Abstract We specify simplicial and categorical constructions used to define the K-theory space for a given exact category. Using the G-construction, we deduce a Grothendieck style presentation for K_1 by generators and relations. Invariants of quadratic forms with values in K-groups in the categorical setting will be also discussed, if time remains. Tuesday, 14 May 2002 No talk currently scheduled Tuesday, 21 May 2002 2:30 - 4:00 Jean-Pierre Marquis Categorical foundations: a brief history (with a philosophical aside) Tuesday, 28 May 2002 2:30 - 4:00 J Lambek An algebraic-computational approach to grammar Tue, 20 August 2002 2:30 - 4:00 Stelios Negrepontis (Athens) The anthyphairetic nature of Plato's dialectics. (Remark: Anthyphaireses = continued fraction) Tuesday, 1 October 2002 3:30 - 5:00 M Barr Epimorphisms in rings of functions Abstract: If X is a topological space (which we always suppose completely regular), then C(X) (resp. C*(X)) denotes the space of continuous (resp. and bounded) real-valued functions on X. We investigate conditions under which a subspace Y of X induces an epimorphism C(Y) ---> C(X). We also show that C*(Y) ---> C*(X) can be epic iff it is surjective which is true only if Y is closed and, in the normal case that is iff. Tuesday, 8 October 2002 2:30 - 4:00 Daniele Bargelli Computational Approach to Arabic Conjugation Tuesday, 22 October 2002 2:30 - 4:00 Mark Weber (U Ottawa) Trees, higher categories and operads Abstract: A distinguishing feature of Michael Batanin's globular approach to higher dimensional algebra, is that trees are regarded properly as combinatorial objects, rather than as certain pointed one-dimensional CW complexes as is more common in algebraic topology. In this talk the role of trees both in higher category theory and in the description of the topological operads important in homotopy theory will be reviewed. Tuesday, 29 October 2002 2:30 - 4:00 Marta Bunge Stack completions revisited ABSTRACT: The notion of a stack over a topos S has been formulated first in terms of a site (Grothendieck-Giraud 1972) and then in terms of the canonical topology of regular epis (Bunge-Pare 1979). Although the latter offers a simpler approach and is meaningful for arbitrary elementary toposes, it is tied up with the question of the representability of stack completions for the category objects (``axiom of stack completions''). The stack completion (for the regular epis) of a (localic) groupoid G in a topos S is the S-indexed category Tors^1(G) of G-torsors. An alternative description, for an etale complete (localic) groupoid G, has been shown [Bunge 1979, 1990] to be describable in terms of the groupoid of (essential) points of the classifying topos BG (``classification theorem'') . In this lecture I will, after discussing briefly the above, (1) first apply the above in order to justify (a refinement of) the assertion made in [Bunge-Moerdijk 1997, after Bunge 1992] that the fundamental groupoid of a locally connected topos E over S is (weakly equivalent to) a prodiscrete groupoid in S, and (2) then discuss a 2-categorical version of the above ``classification theorem'' involving 2-descent, 2-stacks and 2-torsors, and which attempts to explain the role of gerbes, liens and bouquets (Grothendieck, Giraud, Duskin, Breen, Street, Mauri-Tierney) in this context, while pointing to higher-dimensional analogues. Tuesday, 5 November 2002 2:30 - 4:00 Ivan Ivanov Remarks on (oldstyle) Ludics Tuesday, 12 November 2002 2:30 - 4:00 M Barr HSP subcategories of Eilenberg-Moore algebras ABSTRACT Given a triple T on a category C and a factorization system E/M on the category of algebras, we show there is a 1-1 correspondence between full subcategories of the category of algebras that are closed under U-split epimorphisms, products, and M-subobjects and triple morphisms TtoS for which the induced natural transformation between free functors belongs to E. Tuesday, 19 November 2002 2:30 - 4:00 J. Egger (U Ottawa) TBA Tuesday, 26 November 2002 2:30 - 4:00 Ivan Ivanov Remarks on (oldstyle) Ludics II Tuesday, 3 December 2002 2:30 - 4:00 Jean-Pierre Marquis Why Eilenberg & Mac Lane (and others) did not discover adjoint functors and why Kan did Thursday, December 5th, 2002 Lambek Fest (See website for full programme) Tuesday, 10 December 2002 2:30 - 4:00 Mark Weber Analytic 2-functors =================================================== COFFEE: Coffee and cookies will be available after the talk in the lounge. PLACE: BURNSIDE HALL 920, McGILL UNIVERSITY =================================================== (Any comments, suggestions to rags@math.mcgill.ca) Seminar listings are also on the triples WWW page http://www.math.mcgill.ca/triples =================================================== COFFEE: Coffee and cookies will be available after the talk in the lounge. PLACE: BURNSIDE HALL 920, McGILL UNIVERSITY =================================================== (Any comments, suggestions to rags@math.mcgill.ca) Seminar listings are also on the triples WWW page http://www.math.mcgill.ca/triples