Seminars of the CENTRE de RECHERCHE en THEORIE des CATEGORIES CATEGORY THEORY RESEARCH CENTER C ---------> R | / | | / | | / | | / | v / v T ---------> C 30 Jan 2001 2:30 - 5:30 Eduardo Dubuc "Localic Galois Theory I and II" (There will be a half hour break, 3:30 - 4:00 for the usual coffee and cookies, and conversation.) 13 Feb 2001 2:30 - 4:00 R.A.G. Seely A Logic of a Linear Functor 20 Feb 2001 2:30 - 4:00 Eduardo Dubuc (Universidad de Buenos Aires) Localic Galois Theory III (This talk will be self-contained and may be attended by folks who missed the first two parts) 27 Feb 2001 2:30 - 4:00 Jonathan Funk Unramified cosheaf spaces. (Report on joint work with Ed Tymchatyn) Tuesday, 6 March 2001 2:30 - 4:00 Robert Paré Kan extensions for double categories 7 Mar 2001 2:30 - 4:00 Jeff Egger Some little known folklore about Zorn's Lemma This talk, on a WEDNESDAY, will take place in BH 1120 instead of the usual 920. Abstract: This will be the first of a series of talks investigating the Axiom of Choice for Finite Sets [KC] in the context of topos theory. Classically, KC follows from the Linear Ordering Principle, which is a consequence of the Prime Ideal Theorem, which is usually proven using Zorn's Lemma. So it seems appropriate to begin with a reminder of what is known about Zorn's Lemma in toposes, and in particular, how it differs from the Axiom of Choice. Wednesday, 14 March 2001 2:45 - 4:15 Richard Squire Internal maximality in non-localic toposes Abstract: In general, partial orders which are maximal in the external sense are not maximal in the internal sense. We shall show that for a general notion of sketch toposes external will coincide with internal. Location for the Wednesday talks: BH1120 (Tuesday talks remain in BH920) 27 Mar 2001 2:30 - 4:00 Bob Coecke From Birkhoff - von Neumann to Eilenberg - Mac Lane: Categories for the digesting physicist (Abstract on web page) Tuesday, 3 April 2001 2:30 - 4:00 Stephen Watson Transfinite Recursion without the Details Abstract: We describe how many difficult transfinite recursions can be defined by removing the quantifiers of a mathematical instruction by relativizing them according to a schedule to elementary submodels and finite approximations thereof. Our examples come from set-theoretic topology but may be broadly applicable. Tuesday, 10 April 2001 2:30 - 4:00 Richard Wood Decomposing Regularity Abstract: Many important classes of categories are specified by certain types of colimits, certain types of limits, and exactness conditions relating these. If the colimits are given by a KZ-doctrine R and the limits by a co-KZ-doctrine L then it makes sense to enquire about the existence of a distributive law LR--->RL in the sense of Beck. (Essentially, there is at most one such law in this context.) Given such a law, an algebra for the composite doctrine RL is a category C with colimits as prescribed by R, limits as prescribed by L, specification of R-colimits, RC--->C, L-limit-preserving. Examples will be given. The talk will report on joint work with Claudia Centazzo directed towards the problem of solving D=RL, for R, where D is the doctrine for regular categories and L is the doctrine for categories with finite limits. Wednesday, 11 April 2001 2:30 - 4:00 Richard Squire Topos-theoretic characterization of finite-valued presheaves Location: BH920 (Abstract on web page) Tuesday, 17 April 2001 2:30 - 4:00 Bob Coecke From Birkhoff - von Neumann to Eilenberg - Mac Lane: Categories for the digesting physicist II (Abstract on web page) Wednesday, 18 April 2001 2:30 - 4:00 Jeff Egger The L-prime Ideal Theorem, and its Corollaries Abstract: The completeness theorem for coherent propositional logic, alias the Prime Ideal Theorem (PIT) for distributive lattices, is true in certain non-boolean toposes. But it is only strong enough to prove the Order Extension Principle (OEP) for decidable objects. We introduce an extension of coherent propositional logic whose completeness theorem is stronger than (PIT). In particular, it is strong enough to conclude (OEP) for arbitrary objects. This is the second in a series of talks concerning the axiom of choice for (Kuratowski-)finite sets and ``related issues''. Location: BH920 Wednesday, 25 April 2001 2:30 - 4:00 Richard Squire Topos-theoretic characterization of finite-valued presheaves II - "Issues" and a counter-example (Location: BH 920) Wednesday, 2 May 2001 2:30 - 4:00 Jeff Egger Subsingletons and genuine subsingletons (Location: BH 920) Abstract: We introduce the notion of `genuine' subsingleton, which is relevant both to my investigation of Zorn's Lemma and my study of Vermeulen-finitary algebraic structures. Proofs from my last two talks will be improved and/or clarified, and at least one converse will be added. Tuesday, 11 September 2001 2:30 - 3:30 Bob Coecke QUESTIONS on physical logicality (vs. constructivism and resource sensitive provability) 4:00 - 5:00 Boris Ischi Property lattices for separated quantum systems Tuesday, 18 September 2001 2:30 - 4:00 Prakash Panangaden Discrete Quantum Causal Dynamics (Joint work with Rick Blute and Ivan T Ivanov) (Abstract on web page) Tuesday, 25 September 2001 2:30 - 4:00 Prakash Panangaden Discrete Quantum Causal Dynamics II (Joint work with Rick Blute and Ivan T Ivanov) (Abstract on web page) Tuesday, 2 October 2001 2:30 - 4:00 M Barr Flat modules in localic toposes Tuesday, 9 October 2001 CANCELLED 2:30 - 4:00 Alexander Nenashev (U Sask) The theory of quadratic forms in the framework of exact or triangulated categories with duality. Abstract. Such classical notions as quadratic forms, hyper- and metabolic forms, and Witt groups can be introduced and studied over exact or triangulated categories with duality. There are classical invariants of quadratic forms over a field of char different from 2. In low degrees these are the dimension index, the reduced discriminant, and the Hasse-Witt invariant. We discuss a possibility of introducing analogous invariants for quadratic forms over exact categories with duality, with values in suitable subquotients of the higher K-groups. Over triangulated categories with duality, Witt theory has been developed recently by P. Balmer, with interesting applications to the Witt groups of algebraic varieties. Tuesday, 6 November 2001 2:30 - 4:00 Rezaei Siamak From categorial to process grammars Tuesday, 13 November 2001 CANCELLED 3:00 - 4:30 Claudia Casadio Word order and scope in pregroup grammar Tuesday, 4 December 2001 2:30 - 3:30 Bob Coecke Physical realization of the traced monoidal category of Finite dimensional vector spaces. (Joint work with Samson Abramsky) Abstract Exploiting the phenomenon of quantum entanglement (that will be explained) we can operationally realize a setup that mimics the trace construction for the category of finite dimensional vector spaces and as such, via the geometry of interaction construction, the multiplicative fragment of linear logic. This also exhibits the operational difference between the traced monoidal category of sets and partial functions (or relations) either with disjoint union or cartesian product as tensor. 4:00 - 5:00 Claudia Casadio Word order and scope in pregroup grammar Tuesday, 11 December 2001 2:30 - 3:30 Luigi Santocanale Fixed Point Logics and Circular Proofs ABSTRACT: Fixed point logics and $\mu$-calculi are obtained from previously existing logical or algebraic frameworks by the addition of least and greatest fixed point operators. The propositional modal $\mu$-calculus or modal $\mu$-logic, useful for model checking, arises from modal logic exactly in this way. A proposition-term in a $\mu$-calculus has a circular structure: it is possible to travel along subformulas and come back to the starting point using regenerations of fixed point. The circularity is inherited by proof-terms. Normally cut-free proofs in sequent calculi are finite because premiss sequents are always strictly smaller than conclusions. However, in settings where the propositions themselves can be circular, there exists the possibility of having circular or infinite proofs as well. Remarkably, in the theories of fixed points, these kind of proofs happen to be the most useful. Thus, the mathematics suggests that we are indeed allowed to make circular reasonings. In this talk I'll explain this apparent paradox, using the concept of initial algebra (and final coalgebra) of a functor. I'll point out the sense for which initial algebras of functors are a generalization of least fixed points and the relationship to inductively constructed sets (such as lists, finite tress, etc.). I will interpret circular proofs as sort of functions (more precisely, arrows of a category) having as domain an inductively constructed set. Recall that recursively defined functions are uniquely determined by their defining system of equations. Similarly it is possible to transform a circular proof in a system of equations and then prove that this system admit a unique solution. NOTE: Luigi will also be talking at UQAM: Pavillon Président-Kennedy, local PK-4323 Friday December 14 Jouer avec l'induction et la coinduction: les jeux de parité. Un jeu de parité est joué sur un graphe fini de positions et de mouvements. Dans ce graphe on peut bien avoir des cycles, et pourtant on définit l'ensemble des chemins inifinis gagnants pour un des deux joueurs à l'aide de la ``condition de parité''. Cette condition est bien connue dans la théorie des automates qui reconnaissent les objects infinis. En effet tout automate est équivalent à un automate qui utilise la condition de parité. Nous allons définir cette condition et lui donner une interprétation algébrique à l'aide des notions d'algèbre initiale et coalgèbre finale d'un foncteur, c-à-d., à l'aide des formulations catégorielles de l'induction et de la coinduction. Nous allons associer une expression algébrique à chaque jeu de parité et montrer qu'elle dénote (dans la catégorie des ensembles) l'ensemble des stratégies gagnantes pour un des deux joueurs. Nous montrons aussi que les expressions associés aux jeux sont équivalentes aux expressions qu'on peut engendrer par les operations de produit fini, de coproduit fini, d'algèbre initiale et coalgèbre finale, c.à-d aux µ-termes catégoriels. Nous obtenons cette façon une caractérisation explicite de tous les foncteurs qu'on peut définir par des µ-termes dans la catégorie des ensembles. (Apologies for any effects of bad character coding!) =================================================== 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