Seminars of the CENTRE de RECHERCHE en THEORIE des CATEGORIES CATEGORY THEORY RESEARCH CENTER C ---------> R | / | | / | | / | | / | v / v T ---------> C 12 January 1999 2:00 - 3:30 E Schlaepfer (McGill) Another Group Algebra 19 January 1999 2:00 - 3:30 S Crans (McGill) How braidings the post-modern way give rise to old-fashioned braidings, and the failure of this in higher dimensions 4:00 - 5:00 M Makkai (McGill) A duality involving Joyal's disks and omega-categories (joint work with Marek Zawadowski) (Note from rags: this is intended to fill the 4-5 time slot for several weeks, as needed. If anyone else wants to start a regular series later, let me know and we'll arrange it.) (Abstract posted on www site) 26 January 1999 2:00 - 3:30 J Lambek (McGill) Bicategories in algebra and linguistics 4:00 - 5:00 M Makkai (McGill) A duality involving Joyal's disks and omega-categories II 2 February 1999 2:00 - 3:30 C Butz (McGill) Finitely presented Heyting algebras as bi-Heyting algebras ABSTRACT: We show that every finitely presented Heyting algebra is a bi-Heyting algebra, i.e., there is an operation `suplement' defined dual to the notion of implication. Our main technique is algebraic: We use a filtration of a given (f.p.) Heyting algebras by finite distributive lattices. This allows as well to characterize minimal and maximal join-irreducible elements and thus gives yet another proof of the disjunction property of intuitionistic propositional logic. Although not part of this talk it is worthwhile to note that the class of Heyting algebras that are bi-Heyting is much much larger: It includes all (!) free Heyting algebras. References: C. Butz. Finitely presented Heyting algebras. Preprint. Available as http://www.brics.dk/~butz/heyting.ps.gz. S. Ghilardi. Free Heyting algebras as bi-Heyting algebras. C. R. Math. Rep. Acad. Sci. Canada, 14:240-244, 1992. (Proves that Heyting algebras free on a _finite_ set of generators are bi-Heyting.) S. Ghilardi and M. Zawadowski. A sheaf representation and duality for finitely presented Heyting algebras. J. Symbolic Logic 60:911--939, 1995. 4:00 - 5:00 M Makkai (McGill) A duality involving Joyal's disks and omega-categories III 9 February 1999 2:30 - 3:30 M Makkai (McGill) A duality involving Joyal's disks and omega-categories IV 16 February 1999 2:00 - 3:30 V Harnik Significant fragments of classical analysis ABSTRACT: Classical analysis can be developed in a certain strong fragment of second order arithmetic (this fragment is sometimes called, simply, "analysis"). The full strentgh of "analysis" is rarely needed for the most important theorems and a line of research initiated by Friedman, is concerned with the question of determining precisely the subsystem of analysis needed to prove particular theorems. Certain important fragments of analysis have been singled out and their relation to various results of analysis clarified. I will give a survey of the subject and will conclude by outlining a new proof of a theorem of Harrington, concerned with the relationship between two fragments of analysis called "Weak Koenig Lemma" and "Recursive Comprehension Axiom". 4:00 - 5:00 M Makkai A duality involving Joyal's disks and omega-categories V 2 March 1999 2:00 - 3:30 CANCELLED: V Harnik Significant fragments of classical analysis II 4:00 - 5:00 M Makkai A duality involving Joyal's disks and omega-categories V 9 March 1999 2:00 - 3:30 S Crans Yang-Baxter systematically 16 March 1999 2:00 - 3:30 Paddy McCrudden Tannaka duality and quantum groups 23 March 1999 2:00 - 3:30 pm Marta Bunge Distribution Algebras ABSTRACT The purpose of this talk is to report on some recent work on lattice theoretic aspects of S-valued (Lawvere) distributions on an S-bounded topos E. The main result is a duality between the category Dist(E) and the category of S-bicomplete S-atomic Heyting algebras (or "distribution algebras") in E. Closely related with this duality is a relativization of the Pare tripleableness theorem involving a double power set triple. Concretely, we show that Dist(E) is an E-category with a cogenerator o = e^*(Omega_S).1 (where 1 is the terminal S-valued distribution on E), and that the adjoint pair o^(-) -| HOM(-,o): (Dist(E))^op---->E is tripleable. We also examine, in terms of distribution algebras, the passage from distributions on E to complete spreads over E. (This is joint work with Jonathon Funk (UBC), Mamuka Jibladze (Tbilisi) and Thomas Streicher (Darmstadt). 4:00 - 5:00 pm Paddy McCrudden Categories of Representations of Balanced Coalgebroids, Part I. 30 March 1999 2:00 - 3:30 Paddy McCrudden Categories of Representations of Balanced Coalgebroids, Part II 4:00 - 5:00 Sjoerd Crans What's in store in dimension 4 6 April 1999 2:00 - 3:30 RAG Seely Introduction to Linear Bicategories The CRTC has invited the "Higher Dimensional Algebra Seminar" to present (some of) its "deliberations" in the weekly seminar during the spring. We shall begin this week with a talk by M Makkai. Feedback on this arrangement is welcome, of course. 4:00 - 5:00 M Makkai Low dimensional examples of some concepts in higher dimensional categories 13 April 1999 2:00 - 3:30 Carsten Butz Logical aspects of the filter construction. Abstract: The filter construction takes a (regular) category and produces a new one in which all subobject lattices are complete, it may thus be seen as a (functorial) completion. The logic corresponding to these filter categories is regular logic (the fragment of first-order logic containing binary meets and existential quantification) extended by arbitrary meets and the (logical) rule that existential quantification distributes over filtered meets. (Interpretations sound for this logic are a weakening of the notion of saturated models.) More interesting is what happens if one starts with a coherent or a Heyting category. The resulting filter category is again coherent/Heyting and can be canonically embedded into a sheaf topos. Here as well we will describe the corresponding logics. As an application we get for free almost all properties of the non-standard model of arithmetic considered by Moerdijk in "A model for intuitionistic non-standard arithmetic" (Van Dalen Festschrift). The talk is my answer to a discussion I had some weeks ago with Francois Magnan (who studies sub-toposes of the topos of sheaves over a coherent filter category equipped with the finite cover topology) and Gonzalo Reyes. 4:00 - 5:00 Seminar on Higher Dimensional Algebra [ Postponed: 20 April 1999 Sjoerd Crans TBA ] 27 April 1999 2:30 - 4:00 Seminar on Higher Dimensional Algebra 4 May 1999 2:00 - 3:30 Rene Lavendhomme (Louvain-la-Neuve) Structure Symplectique sur un fibre' cotangent en GDS 4:00 - 5:00 Seminar on Higher Dimensional Algebra 11 May 1999 2:00 - 3:30 Carsten Butz Bernays - G"odel Type Theory 4:00 - 5:00 Seminar on Higher Dimensional Algebra 18 May 1999 2:00 - 3:30 Sjoerd Crans Watch a dive in dimension 5 4:00 - 5:00 Seminar on Higher Dimensional Algebra 25 May 1999 2:00 - 3:30 Claudia Casadio Bilinear logic and minimalism 7 Sept 1999 2:30 - 4:00 M Makkai Omega-dimensional homotopy theory 14 Sept 1999 2:30 - 4:00 Luigi Santocanale Free $\mu$-lattices. 21 Sept 1999 2:30 - 4:00 Paddy McCrudden Opmonoidal monads 28 Sept 1999 2:30 - 4:00 Carsten Butz An introduction to Quine's "New Foundations" 5 Oct 1999 1:30 - 3:00 Yoshiki Kinoshita Algebraic explanation of Hoare's notion of refinement 3:30 - 5:00 M Makkai Omega-dimensional homotopy theory II 19 Oct 1999 2:30 - 4:00 D Pavlovic Coinduction over reals and over analytic functions (Abstract on the web page) 26 Oct 1999 2:30 - 4:00 Richard Squire Counting measures: an application of the filter construction to partially ordered algebras 2 Nov 1999 2:30 - 4:00 Mihaly Makkai Universal properties in the higher dimensional context (Abstract on the web page) 9 Nov 1999 2:30 - 4:00 Richard Squire Counting measures: an application of the filter construction to partially ordered algebras II 16 Nov 1999 CANCELLED 2:30 - 4:00 Michael Barr Variations on acyclic models 16 Nov 1999 2:30 - 4:00 Roberto Poli THE UNREASONABLE INEFFECTIVENESS OF MATHEMATICS IN COGNITIVE SCIENCES (Abstract on the web page) 23 Nov 1999 2:30 - 4:00 Sjoerd CRANS Diagrammatic globular sets 7 Dec 1999 2:30 - 4:00 Claudia Casadio About the complexity of the syntactic calculus COFFEE as usual after the first talk PLACE: BURNSIDE HALL 920, McGILL UNIVERSITY =================================================== (Any comments, suggestions to rags@math.mcgill.ca)