July 24, 2001
Richard Wood, The intrinsic Algebra of Dominique Bourn
Abstract: It is hoped that the talk will be accessible to the summer students who have by now heard a little of Category Theory. We will begin with generalities about the point of categorical axiomatics and include some examples before turning to Bourn's programme --- which allows one to isolate many of the arguments of classical modern algebra.

July 31, 2001
Robin Cockett and Robert Seely, Highlights of the first national nano-conference on linear bicategories (Halifax 2001)
Abstract: The two speakers will, in the spirit of a Socratic dialogue, attempt to convey the essential ideas underlying and motivating the development of linear bicategories. The talk should be comprehensible to non-attendees of the nano-conference (and students) - at least for the first half!

Starting from some simple ideas from logic, combinatorics and category theory, we plan to introduce poly-bicategories as a tool for analyzing the notion of modules between (generalized) lax functors.

(Those interested can find some papers on this subject at www.math.mcgill.ca/rags.)

August 14, 2001
Dietmar Schumacher, Extensive Indexed Categories

August 21, 2001
Dale Garraway, Sheaves on an Involutive Quantaloid

August 28, 2001
Dietmar Schumacher, Extensive Indexed Categories II

September 11, 2001
Bob Paré, Geometric Patterns Arising from the Adjunction of Adjoints
Abstract: The 2-cells in the 2-category obtained from a graph by taking the free category and then formally adding right adjoints for each morphism can be described in terms of certain diagrams which resemble Kauffman diagrams or Feynman diagrams. We shall examine some examples of this.

September 18, 2001
Richard Wood, Tensor Products of Sup Lattice

September 25, 2001
Bob Rosebrugh, Minimal Realization 3.0
Abstract: The context is the program studying the bicategory of spans of graphs as an algebra of processes with applications to concurrency theory. This talk describes a study of functorial aspects of reachability, minimization and minimal realization. The compositionality of minimization has a striking application to model-checking. (This is joint work with N. Sabadini and R.F.C. Walters)

October 30, 2001
Richard Wood, V-indexed categories redux
Abstract: For V a monoidal category, the usual 2-category V-\cat is deficient in a variety of ways that stem from the heterogeneous nature of its objects. More precisely, since a V-category has objects in set and arrows in V, the important Grothendieck constructions of cat (= set-cat) --- that build new categories with objects of the new using arrows of the old --- have a tendency to be weaker when imported to V-cat. This problem, which was addressed in the speaker's recent Ph D thesis, will be revisited with the help of the still more recent concept of enrichment in a bicategory. The material is inherently technical but if audience demand so warrants the presentation will begin with an account of classical enrichment and be spread over several talks. Joint work in progress with Robin Cockett.

November 13, 2001
David Lever, A Neurological Study Of Mathematics
Abstract: The adjointness properties of universal and existential quantification in Sets leads to a neural network description of mathematics whereby lower and upper bounds on the truth of mathematical statements can be calculated and used to prove things. Although we do not claim that the new neural networked language of mathematics is better than the classical language, we do hope it will lead to one more computable.

November 20, 2001
Richard Wood, More on pro-W-categories
Abstract: This talk will continue the earlier ones. Recall that, for W a bicategory, we had defined a 2-category pro-W-cat and shown it to be quite robust with respect to the usual bicategorical constructs over W. We will begin this talk with examination of special cases --- for example, that where W has but one object and that where W is merely a category --- in an attempt to relate this work to that of others in our group. Still with the `vertical' view in mind we will also define pro-W-profunctors leading to the unfortunately named bicategory pro-W-pro whose `maps' essentially recover pro-W-cat --- modulo Cauchy completeness. We will move on to a `horizontal' view of pro-W-cat and pro-W-pro (although as of this writing it would be disingenuous to say that this later work is `in press') and establish a biequivalence between the views introducing the audience to still further bicategorical constructs. It is our thesis that this biequivalence is `indexing' and that it incorporates within it both enrichment and variation as important special cases (neither of which is seen adequately by imposing it within the confines of the other) --- each with a vertical and a horizontal view.

November 27, 2001
Luzius Grunenfelder, On braided and ordinary Hopf algebras
Abstract: After a brief review of some highlights in the structure theory of ordinary Hopf algebras we will see how certain braided Hopf algebras occur naturally in this theory. They can be understood as the "infinitesimal part" of ordinary pointed Hopf algebras, very much like the Lie algebra (or its universal envelope) of a Lie group is the infinitesimal part of that group. We will see how such braided Hopf algebras can be constructed, and explore some ways to construct ordinary Hopf algebras from braided ones.

January 8, 2002
Luzius Grunenfelder, On braided and ordinary Hopf algebras II
Abstract: A continuation of previous talk.

January 15, 2002
Richard Wood, Tax Everything!

January 29, 2002
Peter Schoch, Propositions Suck [sic] (concerning the future of logic as we know it)

February 26, 2002
Bob Paré, Free Adjoints vs Spans

April 9, 2002
Richard Wood, A new approach to certain duality theorems

April 16, 2002
Mitja Mastnak, Hopf Algebra Extensions Arising From Semi-Direct Products of Groups
Abstract: We study Hopf algebra extensions arising from semi-direct products of groups in terms of group cohomology. This enables us to compute and describe explicitly some groups of Hopf algebra extensions.

April 23, 2002
David Lever, In Defense Of Skolem Functions
Abstract: A Skolem function is a term with special purpose. We will use Skolem functions to interpret first order theories in indexed categories (with reasonable properties possessed by Sets, Sheaves, Cat, etc.). The Skolemizations allow indexed simplex tableaus to be attached to theories with linear optimization providing proofs. The connection of a dual simplex to an originating theory is still not understood.

April 30, 2002
Dietmar Schumacher, Cat(S) is cartesian closed