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*