Tuesday, July 4, 2006
Elango Panchadcharam, Mackey Functors and Green Functors

Tuesday, July 4, 2006
Paul Taylor, Induction/Recursion

Tuesday, July 11, 2006
Andrew Baker, Minimal atomic objects in algebraic and topological derived categories
Abstract: In joint work with J.P. May and R. Pereira we studied notions of minmal atomic and irreducible objects in the homotopy category of finite type p-local CW spectra. This work was extended by my student M. Alshumrani to the algebraic derived category of finite type complexes over a commutative Noetherian local ring. I will describe these results, and mention some outstanding problems and further possible generalizations.

Tuesday, August 8, 2006
Mitja Mastnak, On Bimeasurings
Abstract: If $A$ and $B$ are algebras and $C$ is a coalgebra, then a linear map $\psi\colon B\ot C\to A$ is called a measuring if it induces an algebra map from $B$ to the convolution algebra $Hom_k(C,A)$. There is a universal measuring coalgebra $M(B,A)$ and measuring $\theta\colon M(B,A)\ot B\to A$ for every pair of algebras $A$ and $B$ such that $C$-measurings from $B$ to $A$ correspond bijectively to coalgebra maps from $C$ to $M(B,A)$. If $B$ and $C$ are bialgebras, then we call a map $\psi\colon B\ot C\to A$ a bimeasuring, if it measures in both variables. If $A$ is commutative then $M(B,A)$ carries a natural bialgebra structure making it a universal bimeasuring bialgebra. In the talk I will explore (monoindal) properties of the universal bimeasuring bialgebra functor $B(-,A)\colon Bialg\to Bialg$. In particular, I will show that there is a natural transformation $\alpha\colon B(-,A)\otimes B(-,A)\to B(-\otimes -,A)$ satisfying $\alpha(1\otimes \alpha)=\alpha(\alpha\otimes 1)$ and that no natural transformation $\beta\colon B(-\otimes -,A)\to B(-,A)\otimes B(-,A)$ satisfying $(\beta\otimes 1)\beta=(1\otimes \beta)\beta$ exists. (Joint work with L. Grunenfelder)

Tuesday, August 15, 2006
Francisco Marmolejo, CCD lattices in set^(C^op)
Abstract: We consider a characterization of constructively compltely distributive (CCD) lattices in set^(C^op) along the lines of the characterizaton of sup lattices in the classical work of Joyal and Tierney: "An extension of the Galois theory of Grothendieck". So a CCD lattice L in set^(C^op) will be a functor L:C^op--->ccd(set) with additional properties.

Tuesday, September 12, 2006
Bob Paré, Introduction to Double categories
Abstract: I will review the definition if double category, give some examples and discuss the relationship with other concepts of two dimensional categories.

Tuesday, September 19, 2006
Sam Howse, NummSquared: a new well-founded functional foundation for formal methods
Abstract: Set theory is the standard foundation for mathematics, but often does not include rules of reduction for function calls. Therefore, for computer science, the untyped lambda calculus or type theory is usually preferred. The untyped lambda calculus (and several improvements on it) make functions fundamental, but suffer from non-terminating reductions and have partially non-classical logics. Type theory is a good foundation for logic, mathematics and computer science, except that, by making both types and functions fundamental, it is more complex than either set theory or the untyped lambda calculus. This talk proposes a new foundational formal language called NummSquared that makes only functions fundamental, while simultaneously ensuring that reduction terminates, having a classical logic, and attempting to follow set theory as much as possible. NummSquared builds on earlier works by John von Neumann in 1925 and Roger Bishop Jones in 1998 that have perhaps not received sufficient attention in computer science. Usual set theory, the work of Jones, and NummSquared are all well-founded. NummSquared improves upon the works of von Neumann and Jones by having reduction and proof, by supporting computation and reflection, and by having an interpreter called NsGo (work in progress) so the language can be practically used. NummSquared is variable-free. For enhanced reliability, NsGo is an F\#/C\# .NET assembly that is mostly automatically extracted from a program of the Coq proof assistant. As a possible step toward making formal methods appealing to a wider audience, NummSquared minimizes constraints on the logician, mathematician or programmer. Because of coercion, there are no types, and functions are defined and called without proof, yet reduction terminates. NummSquared supports proofs as desired, but not required. My thesis on NummSquared may be found at: http://nummist.com/poohbist/index.html

Tuesday, September 26, 2006
Tobey Kenney, Introduction to Copower Objects

Tuesday, October 3, 2006
Benoit Valiron, On a fully abstract model for a quantum linear lambda calculus
Abstract: We study the linear fragment of the quantum lambda calculus, a programming language for quantum computation with classical control that was described in (Selinger, Valiron, 2006). We sketch the language and discuss a categorical model. We also describe a fully abstract denotational semantics in the category of completely positive maps.

Tuesday, October 10, 2006
Bob Rosebrugh, Sketches and categorical database design (with implementation system demo)
Abstract: Finite-limit, finite-sum (EA) sketches are the best syntactic structure for modelling databases and their views'. With Michael Johnson, RJ Wood and others, we have explored and exploited this observation about EA sketches and called it the Sketch Data Model (SkDM). The model extends and enhances the standard ERA data model. In particular, it provides a context for the problems of updating views and database integration. We will begin with an overview of this work. Using Java to provide portability, students at Mount Allison and I have written an application that provides a user-friendly graphical design environment for EA sketches, allows saving a design into an XML document and exporting that to a database schema in SQL (the standard relational database language). The application and some of its capabilities will be demonstrated.

Tuesday, October 17, 2006
Geoff Cruttwell, Normed and Ordered Algebraic Structures
Abstract: ABSTRACT In this talk, I'll investigate a number of interesting similarities between algebraic structures with a norm, and algebraic structures with an order. The similarities will be highlighted through the identification of both normed and ordered abelian groups as examples of lax monoidal functors.

Tuesday, October 24, 2006
Toby Kenney, The Lattice Structure on QX

Tuesday, October 31, 2006
Bob Paré, Introduction to Profunctors
Abstract: As the title says, this will be an introduction to the notion of profunctor, the two dimensional version of relation. This is all well-known stuff or at least should be. Unless, of course, I can think of something new before Tuesday.

Tuesday, November 14, 2006
Peter Selinger, How to use category theory to write a compiler I
Abstract: It is not every day that one has a nice story to tell about a connection between two apparently unrelated subjects. In this talk, I will describe a sequence of ideas that leads from a construction in category theory, via lambda calculus and Krivine's abstract machine, to an actual compiler for a programming language. I realize that some people consider category theory "too abstract", while some others consider compilers "too applied". I hope that this talk will appeal to the union, rather than the intersection, of these two groups of people.

I'll start by reviewing the connection between lambda calculus and cartesian-closed categories, then take it slowly from there. I will explain the concept of a "continuation", and continuation passing style (CPS) translations. I may not get much further this week, so there is the possibility of a sequel.

Tuesday, November 21, 2006
Peter Selinger, How to use category theory to write a compiler II
Abstract: This is a continuation of last week's talk. I will start by explaining the CPS translation more carefully - while it is easy to understand from a categorical point of view, we yet have to gain some insight into its computational interpretation. This will lead us naturally to Krivine's abstract machine.

Tuesday, November 28, 2006
Peter Selinger, How to use category theory to write a compiler III
Abstract: Last week, we had a detailed look at the CPS translation for call-by-name lambda calculus, which was determined by a categorical construction. In the third (and final) installment of this mini-series, I will describe how this leads to Krivine's abstract machine, which can in turns be directly compiled to assembly code.

Tuesday, December 5, 2006
Heulwin Rankin, Coalgebras, corings and their comodules
MSc presentation

Tuesday, January 9, 2007
Richard Wood, A bicategory with binary products and a terminal object is' a monoidal bicategory
Abstract: Products in a bicategory are defined via birepresentability. Accordingly, their universal property is considerably weaker than that of a product, in the usual sense, in a 2-category.

A monoidal bicategory is by definition a one-object tricategory. As such, the pentagon condition' for the associativity e q u i v a l e n c e, \alpha, is replaced by an invertible modification, \pi, which is required to satisfy, amongst other things, Stasheff's nonabelian 4-cocycle condition. It is an equality of a pasting of 5 particular 2-cells with a pasting of 4 other 2-cells, all involving \alpha and \pi. If drawn on the surface of a sphere, the diagram has 14 vertices and 21 edges bounding the 9 2-cells.

Thus, to show that a bicategory with finite products is' a monoidal bicategory with binary product as tensor requires the construction of such data as \alpha and \pi, verification of various pseudonaturalities etcetera, and satsisfaction of conditions, including Stasheff's --- all from the rather weak universality of products in a bicategory. I will report on a beautiful proof of this theorem, given by Max Kelly, based on an idea he attributes to Ross Street, which does not involve a n y diagrams. The talk should be accessible to anyone who knows what an equivalence of mere categories is and what products in mere categories are; and who is furthermore willing to relentlessly combine these ideas.

Tuesday, January 16, 2007
Renzo Piccinini, Some representable topological functors
Abstract: In what follows $\set$ (resp. $\set_*$) stands for the category of sets and functions of sets (resp. of based sets and based functions of sets). Now let $\ci$ (resp. $\ci_*$) be a subcategory of the category $\tp$ of topological spaces and maps and let $H\ci$ be the homotopy category associated to $\ci$.

We say that a (contravariant) functor $$F:H\ci\rightarrow \set$$ is {\em representable} if there exists a space $Y$ (possibly in a category $\ci'$ larger than $\ci$) and a natural equivalence $\eta:[-,Y]\to F$; the object $Y$ is said to be a {\em classifying space} for $F$.

We begin by giving some examples of representable topological functors and the classifying spaces associated to them. One of the best known representable functors is the following: let $G$ be a topological group and let $\xi_G:CW\to \set$ be the functor which takes any CW-complex $B$ into the set of all equivalence classes of principal $G$-bundles over $B$; this allows us to classify principal $G$-bundles. Due to its local triviality a principal $G$-bundle over a CW-complex is a (Hurewicz) fibration; we shall address the problem of classifying fibrations.

Tuesday, January 23, 2007
Toby Kenney, The Axiom of my Choice

Tuesday, January 30, 2007
Richard Wood, More on Profunctors

Tuesday, February 6, 2007
Toby Kenney, Russell Finiteness I

Tuesday, February 13, 2007
Toby Kenney, Russell Finiteness II

Tuesday, February 27, 2007
Bob Paré, Some Thoughts on Coproducts

Tuesday, March 6, 2007
Bob Rosebrugh, Updating the database view update problem
Abstract: The concept of view' is an important element of data modelling. The view update problem': "when can a change in a view state be lifted to a total database state?" has been the subject of ongoing research in the relational and other data models. The literature on the view update problem mostly dates to the 1980's, but there has been a recent revival of interest from the database community.

The sketch data model (SkDM) provides clear categorical criteria determining when there is an optimal (i.e. universal) solution to this problem. The talk will review the literature and include a decription of the how the SkDM solution reformulates and extends this work.

Tuesday, March 13, 2007
Toby Kenny, Finiteness via Quotient Objects

Tuesday, March 20, 2007
Peter Selinger, Introduction to traced monoidal categories

Tuesday, March 27, 2007
Peter Selinger, Introduction to traced monoidal categories: Examples and applications

Tuesday, April 3, 2007
Benoit Valiron, Random walk in categorical models for intuitionistic logic and higher-order quantum computation

Tuesday, April 10, 2007
Richard Wood, A bicategory with finite products is a one-object, one-arrow, one 2-cell, one 3-cell weak six-category, that is a symmetric monoidal bicategory

Tuesday, April 17, 2007
Toby Kenney, Generating families and well-piontedness

Tuesday, April 24, 2007
Margaret Beattie, Coquasitriangular Hopf algebras and categories of Yetter-Drinfeld modules - Part I
Abstract: This talk will give some background on quasitriangular Hopf algebras and the Drinfeld double, and will then introduce the notions of the generalized double and of coquasitriangular Hopf algebra. We will also introduce the category of left Yetter Drinfeld modules over a Hopf algebra, and the notion of a Hopf algebra in this category. If time permits, we'll give necessary and sufficient conditions for a generalized double to be a biproduct in the sense of Radford, so that a braided Hopf algebra naturally appears.

Tuesday, May 1, 2007
Daniel Bulacu, Coquasitriangular Hopf algebras and categories of Yetter-Drinfeld modules - Part II

Tuesday, May 8, 2007,
Eric Paquette, Some quantum categorical questions (QCQ): Beyond the "shut-up and calculate" paradigm