Tuesday, July 15, 2008
Micah McCurdy, Cyclic Structures in *-autonomous Categories
Abstract: David Yetter, working with quantales, introduced the notion of a _cyclic element_, namely, an element z such that x @ y <= z if an only if y @ x <= z for all y and x. Generalizing this, various authors (among them Kimmo Rosenthal; as well as Blute, Ruet, and Lamarche) have considered monoidal categories with cyclic elements, of particular interest is when the monoidal categories are star autonomous and the dualizing element is cyclic, although there is some confusion as to the relevant coherence conditions in this case. The presentation of star-autonomous categories as linearly distributive categories with negation sheds insight into the structural isomorphisms present in a star-autonomous category, and motivates possible coherence axioms for any extra structure. We give an intuitive set of axioms under which we characterize cyclic structures on braided star-autonomous categories as equivalent to twists for the braiding.

In particular, Rosenthal's result concerning endoprofunctors can be clarified and put into this light, where again the same axioms crop up.

Tuesday, September 9, 2008
Richard Wood, Lax Duals
Abstract: Recall that a {\em bidual situation} in a monoidal bicategory $({\cal M},\otimes, I,...)$ consists of data $(A,B;n,e;\eta,\epsilon)$, where $A$ and $B$ are objects of ${\cal M}$, $n:I\to B\otimes A$ and $e:A\otimes B\to I$ are arrows (called unit and counit respectively), and (collapsing \otimes to concatenation) $\eta:1_B\to Be.nB$ and $\epsilon:eA.An\to 1_A$ are invertible 2-cells (called constraints) satisfying two equations.

A bidual situation in $({\cal M},\otimes, I,...)$ is the same thing as a pseudoadjunction in the one-object suspension tricategory. We say that $B$ is a right bidual of $A$ and so on. In the monoidal bicategory $({\rm prof},\times, 1,...)$, each category $A$ has a right bidual provided by $A^{\rm op}$.

A lax adjunction in a tricategory, in particular a lax dual situation in monoidal bicategory consists of data $(A,B;n,e;\eta,\epsilon)$ as above except that $\eta$ and $\epsilon$ are not required to be invertible. In our paper ``Duals Invert'' we proved three theorems about lax duals which lend credence to the further slogans ``Lax Duals Adjoin'' and ``Lax Transport Exists''. The talk will be centred on these theorems.

Tuesday, September 16, 2008
Bob Paré, Transport of Structure along Adjoints
Abstract: I will consider various aspects of how monoidal structures can be transported from one category to another along an adjoint pair. This is a preliminary talk who's only virtue may be to generate discussion.

Tuesday, September 23, 2008
Gastón Andrés García (Córdoba), Hopf algebras and quantum groups
Abstract: Hopf algebras were introduced in the 50's and they have been studied systematically since the 60's, first related with algebraic groups and later as objects of interest in themselves.

In this talk we will discuss the problem of classifying finite-dimensional Hopf algebras over an algebraically closed field of characteristic zero. One of the main obstructions to solving the problem is the lack of enough examples. In the last part of the talk, we will show how to construct new examples from quantum groups using a characterization of the quantum subgroups of a simple quantum group at a root of unity.

Tuesday, September 30, 2008
Geoff Cruttwell, Normed spaces and the change of base for enriched categories (Ph.D. internal defence)
Abstract: In this thesis, we study two related concepts: a generalization of the concept of normed space to a categorical setting, and a study of the change of base for enriched categories. After briefly describing the first idea, we will show how it leads to a desire to further understand the change of base. This, in turn, leads to an interesting comparison between bicategories and (pseudo) double categories.

Tuesday, November 4, 2008
Phil Scott (University of Ottawa), Geometry of Interaction
Abstract: ABSTRACT

Tuesday, November 18, 2008
Avra Laarakker (Acadia University), Topological Properties of Radix Representations for Vectors
Abstract: My research project involves the study of digital representations for vectors. The most common digital representations are radix representations, such as the standard base 10 representation for real numbers. A number, such as 567, has ones (7), tens (6), and hundreds (5) digits, and so on. This means that 567 is equal to 7 times 1, plus 6 times 10, plus 5 times 100. Each digit corresponds to a power of 10, thus 10 is called the base, or radix. The digits are chosen from a given digit set, such as {0, 1, ..., 9}.

There are many digital representations for real numbers, including continued fractions, beta transformations, Luroth series, and generalizations of these. In higher dimensions we have digital representations for vectors, including multidimensional continued fraction algorithms.

My supervisor, Dr. Eva Curry, has also developed radix representations for vectors with integer entries. Here, the base is a square matrix with integer entries, and the digits are vectors. The inter-relations between different representations give important number theoretic and topological results about the behavior of certain transformations on vectors.

Studying the digital representation of vectors will yield information about the topological properties of the set of "fractions". My goal is to determine the relationship between the dilation matrix and digit set used and topological properties of the resulting self-affine tile, such as connectedness or whether the tile is disc-like (topologically equivalent to a disc).

In my presentation, I will summarize previous important results leading up to and which are relevant to our problem as well as outline recent results.

Tuesday, January 6, 2009
Mitja Mastnak (Saint Mary's University), Free probability and Hopf algebras
Abstract: I will try to explain how combinatorial Hopf algebras can be used to study joint distributions of k-tuples in a noncommutative probability space. In recent joint work with A. Nica we have constructed a Hopf algebra whose multiplication of characters corresponds to free multiplicative convolution of joint distributions. I will highlight the case k=1 when the combinatorial Hopf algebra in question is the well known Hopf algebra of symmetric functions. In this case several notions in free probability, such as the S-transform, its reciprocal 1/S, and its logarithm log S, relate in a natural sense to the sequences of complete, elementary and power sum symmetric functions.

Tuesday, January 13, 2009
Bob Paré, Embedding the double category of categories into quantals
Abstract: Given a small category, the set of subsets of arrows forms a quantale. This is one way of making the statement that a category is a monoid except that multiplication is not always defined precise. We will examin varius aspects of this construction. This is joint work with Toby Kenney.

Tuesday, March 10, 2009
Geoff Cruttwell, The Mon Construction
Abstract: The two immediate generalizations of category - internal categories, and enriched categories - are united by the idea of monads in a bicategory. That is, an internal category in C is a monad in the bicategory Span(C), and a V-category is a monad in the bicategory V-Mat. However, in relation to these constructions, taking monads in a bicategory is not as good as one might hope. In particular, morphisms of monads give neither internal functors in the first case, or V-functors in the second (or internal profunctors or V-profunctors, for that matter). Another problem is that Mon is not an endo-2-functor on bicategories - for the simple reason that bicategories, lax functors, and lax (or op-lax) natural transformations don't form a 2-category!

However, by moving to double categories, we can resolve the functor problem, as well as show that Mon is a 2-functor. This is a beautiful construction, and deserves to be more widely known.

Tuesday, March 17, 2009
Bob Paré, The Structure of Spans
Abstract: This is joint work with Robert Dawson and Dorette Pronk in which we study the functorial properties of the span construction.

Tuesday, March 31, 2009
Bob Rosebrugh, Algebras, update strategies and fibrations
Abstract: For a category C with products and an object V, the sum functor on the slice category C/V sending X --> V to X is a left adjoint. Algebras for the generated monad are what have been called `lenses' when C = Set. Lenses have appeared at least three times in theoretical Computer Science, most recently as `update strategies' for database views. In sets, lenses are the `constant complement' update strategies studied in the 1980's. For ordered sets as base, the more recent work of S. Hegner is recovered. Lenses in Cat make sense too, and we can now relate them to the (op)fibration criterion for updatability.

This is joint work with Michael Johnson and Richard Wood

Tuesday, April 7, 2009
Bob Rosebrugh, Algebras, update strategies and fibrations (continued*)
Abstract: For a category C with products and an object V, the sum functor on the slice category C/V sending X --> V to X is a left adjoint. Algebras for the generated monad are what have been called `lenses' when C = Set. Lenses have appeared at least three times in theoretical Computer Science, most recently as `update strategies' for database views. In sets, lenses are the `constant complement' update strategies studied in the 1980's. For ordered sets as base, the more recent work of S. Hegner is recovered. Lenses in Cat make sense too, and we can now relate them to the (op)fibration criterion for updatability.

* A short review of the first part may be given for those who missed the first part. RP

This is joint work with Michael Johnson and Richard Wood

Tuesday, May 12, 2009
Dorette Pronk, Orbifold Atlas Groupoids
Abstract: In this talk I will revisit the construction of a smooth groupoid from an orbifold atlas. I will show that this construction can be viewed as an internal version of a category of fractions in the sense of Gabriel and Zisman. I will then discuss what is required of an atlas for a generalized smooth structure to allow for a smooth groupoid representation.

Tuesday, June 2, 2009
Gabor Lukacs, On quasi-convexity and the Pontryagin dual of products
Abstract: While products of topological groups are easy to describe (they are just the product space with coordinate-wise operations), their coproduct is rather complicated even in the abelian case. What complicates the situation even more is that the Pontryagin dual of a product of locally compact abelian groups is not the coproduct of the duals topologically -- at least not when we are considering at the category of all abelian topological groups. The answer is hidden in the so-called "asterisk" topology, which turns out to be the coproduct in the category of the locally quasi-convex abelian topological groups.

In this talk, a thorough introduction to locally quasi-convex groups and the Pontryagin dual will be presented. It will be followed by a survey of results relating duals of products with coproducts.