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
*