Geoff Cruttwell,

*A Study of CCD Lattices in a Functor Category*

**Abstract**: In their monograph ``An Extension of the Galois Theory of Grothendieck'', Joyal and Tierney characterized sup lattices and locales in a functor category set^C^op. In the early 1990's, Fawcett and Wood introduced a new lattice concept which is a special case of locales: constructively completely distributive (CCD) lattices. This work brings together these two ideas by attempting to characterize CCD lattices in a functor category.

Tuesday, September 13, 2005

Francisco Marmolejo, *Locale morphisms with exact direct image functor in sheaves*

**Abstract**:
The ultraproduct funtor set^I---> set, for a set I and U an ultrafilter
on I can be recorverd up to isomorphism as the composite

set^I--->sh(I)--->sh(beta(I))--->set

where beta(I) is the Stone-Cech compatification, the second map is induced
by the usual embbeding I--->beta(I) and the third is induced by choosing
the ultrafilter U, 1--->beta(I). The reason this map preserves models for
regular or exact or pretopos categories is that sh(I)--->sh(beta(I)) does.
In my Ph.D. thesis there is a characterization of such maps in
topological spaces, that is, those that induce exact direct image functors
in sheaves. An extension of this characterization for locales will be the
subject of this talk.

Tuesday, September 20, 2005

Bob Paré, *NEWS FLASH: Beck Condition Deemed Irrelevant*

**Abstract**:
A group of researchers in Halifax, Canada (Robert Dawson, Dorette Pronk
and Robert Paré of RDP Associates) have discovered a theory which allows
them to dispense with the Beck condition, long thought to be an essential
feature of the span construction. Now spans can be used in situations
never before thought possible. Learn how and grow your research.

Tuesday, September 27, 2005

Bob Paré, *Span, Span, Span, ...*

**Abstract**:
In this continuation of last week's talk, I will introduce
various Span constructions for categories without pullbacks and for
2-categories as well. Some specific examples will be examined and
applications given.

Tuesday, October 11, 2005

Jeff Egger, *Tutorial" on *-Autonomous Categories*

**Abstract**:
What are *-autonomous categories, and why might one care
about them, let alone try to generalise a known technique for their
construction? These are the questions which I will try to answer
during this talk (with, I suspect, varying degrees of success). If
time permits, I will also discuss a generalisation (due to me) of a
technique for constructing *-autonomous categories (due to Schalk and
de Paiva).

Tuesday, October 18, 2005

Georg Hofmann, *Coxeter Groups: Groups Generated by Reflections*

**Abstract**:
Coxeter groups play an important role in many areas of geometry and
algebra, for example in the classification of regular polyhedra in
euclidian or hyperbolic space or in the classification of the
finite-dimensional, simple Lie algebras. I propose to present an
introduction to some basic Coxeter Theory and to report on
a characterization of Coxeter groups as groups generated by
reflections on a graph. This recent result is a useful tool
for the investigation of geometric group actions
generated by reflections.

Tuesday, October 25, 2005

Jeff Egger, *Of operator algebras and operator spaces*

**Abstract**:
One of the recent advances in Functional Analysis has been
the introduction of the notion of an (abstract) operator space.
This can be seen as a refinement of the notion of a Banach space which
(among other things) solves the problem that not every Banach algebra
is an operator algebra. Which theorems about Banach spaces generalise
to operator spaces? This question would be easier to answer if one
could prove Pestov's Conjecture: that there exists a Grothendieck
topos whose internal Banach spaces are equivalent to operator spaces.
I will report on progress towards proving Pestov's conjecture.

Tuesday, November 1, 2005

Gavin Seal, *Lax algebras and the Kleisli category*

**Abstract**:
To complete the previous talk, we will detail an essential problem in
the construction of lax algebras, and explain how a certain
Kleisli-related construction may be used to solve it.

Tuesday, November 15, 2005

Richard Wood, *Cartesian Bicategories II*

**Abstract**:
The notion of ` cartesian bicategory', introduced by Carboni and
Walters for locally ordered bicategories, is extended to general
bicategories. Bicategories of spans will be characterized as
cartesian bicategories in which every object is discrete, every
comonad has an Eilenberg-Moore object, and for every object $X$,
the left adjoint arrow $X\ra I$, where $I$ is terminal with respect
to left adjoints, is comonadic. Bicategories of relations will be
revisited from the present point of view while the full generality
will also be used to characterize bicategories of internal categories
and internal profunctors with respect to a suitable base.

(This joint work by A. Carboni, G.M. Kelly, R.F.C. Walters and R.J. Wood was reported in the @CAT seminar before many new people joined us. An attempt will be made both to keep the talk self-contained and add some new material for continuing members of the seminar.)

Tuesday, November 22, 2005

Kia Dalili, *The Number of Generators of Hom_R(A,B)*

**Abstract**:
Given two finitely generated R-modules A and B, what
can we say about the number of generator of the R-module \Hom_R(A,B).
This talk will be a report on the use of cohomological
degree functions in bounding the number of generators of Hom_R(A,B).
I will discuss several special cases and present some evidence for
the existence of a polynomial bound in terms of certain degree
functions.

Tuesday, November 29, 2005

Peter Selinger, *Control categories, classical logic, and control operators*

**Abstract**:
There is a three-way correspondence between category theory,
logic, and programming languages. The most well-known instance of this
correspondence relates cartesian-closed categories, intuitionistic
propositional logic, and simply-typed lambda calculus. I will briefly
review some of the above-mentioned items, and then show how to extend
the correspondence to classical logic. On the programming language
side, one obtains Parigot's lambda mu calculus, which is a language
with certain control operators (they allow the evaluation of an
expression to be interrupted). I will describe a class of categorical
models for this language called "control categories". They are based
on Power and Robinson's premonoidal categories. I will show that the
call-by-name lambda mu calculus forms an internal language for control
categories. Moreover, the call-by-value lambda mu calculus forms an
internal language for the dual co-control categories. As a
consequence, one obtains a beautiful categorical semantics of
classical logic, and a remarkable duality between call-by-name and
call-by-value programming languages.

Tuesday, December 6, 2005

Peter Selinger *Control categories and duality (continued)
*

**Abstract**:
Last week, I described the connection between simply-typed
lambda calculus and intuitionistic propositional logic. I also gave an
indication of how the connection could be extended to classical logic,
resulting in Parigot's lambda-mu calculus. I pretty much ignored
category theory.
This week, I'll focus on a class of categorical models for classical
logic, called "control categories". They are based on Power and
Robinson's premonoidal categories. I plan to show that the
call-by-name lambda mu calculus forms an internal language for control
categories. Moreover, the call-by-value lambda mu calculus forms an
internal language for the dual co-control categories. As a
consequence, one obtains a remarkable duality between call-by-name and
call-by-value programming languages, in the presence of control
operators.

Tuesday, January 10, 2006

Richard Wood, *Discreteness and Tabulation in Cartesian Bicategories
(Further report on joint work with Carboni, Kelly, and Walters)
*

**Abstract**:
A BCDE is a

Bicategory which is

Cartesian in which every object is

Discrete and in which every comonad has an

Eilenberg-Moore object and every map is comonadic.

Previous talks in this series have dealt with axioms B and C. This
talk will begin with a treatment of {\em groupoidal} objects, those
X for which the canonical 2-cell

d\ox X X\ox X-------->X\ox X\ox X | | d^*| delta |X\ox d^* | ======> | v v X------------->X\ox X dis invertible and {\em ordal} objects, those X for which the unit 1_X---->d^*d is invertible. Algebraists will recognize groupoidal as a formulation of separabilty for certain algebras. Categorists will recognize groupoidal for X as implying X-|X with respect to \ox regarded as a composition. {\em Discrete} objects are those which are both groupoidal and ordal. It is claimed that axioms C, D, and E characterize bicategories of the form spanE, for E a category with finite limits.

Tuesday, January 17, 2006

Jeff Eggar, *Recollections of Schenectady*

Tuesday, January 24, 2006

Sara Faridi, *Algebra using simplicial complexes*

**Abstract**:
In this talk we show how one can associate an ideal to a simplicial
complex (or a hyper-graph), and use combinatorial properties of
simplicial complexes to deduce algebraic properties of the
associated ideal. One can use this approach to find classes of
complexes with certain algebraic properties. In particular, motivated
by graph theory, we show how one can define structures such as
simplicial tree, simplicial cycle and chordal complex whose
corresponding ideals have nice algebraic properties. The talk will be
survey-style: all are welcome!

Tuesday, January 31, 2006

Sara Faridi, *Algebra using simplicial complexes (continued)*

Tuesday, February 14, 2006

Gabor Lukacs, *Introduction to duality of abelian groups I*

Tuesday, February 28, 2006

Gabor Lukacs, *Introduction to duality of abelian groups II*

Tuesday, March 7, 2006

Richard Wood, *Cartesian Bicategories II (continued) Beck-Chevalley conditions*

**Abstract**:
When I last reported on this work I began with the gratuitously contrived
`A BCDE is a ...'. I had earlier reported on the basics of what it means
for a Bicategory to be Cartesian and skipped `every object is Discrete'
in favour of telling you about `every comonad has an Eilenberg-moore object
and every map is comonadic'. At that point all that remained to prove that
A BCDE is a bicategory of the form SpanE, where E is a category with finite
limits, were a couple of Beck-Chevalley conditions. Since then I have proved:
A BCE satisfies those Beck-Chevalley conditions a n d `every object is
Discrete'.

So the theorem is better although some would like axioms that include D and less E, for notice that RelE with E regular does not enjoy `every map is comonadic'. Time permitting, I'll address that issue too.

Tuesday, March 14, 2006

Gabor Lukacs, *Pro-C^*-algebras: Non-commutative k-spaces*

**Abstract**:
A pro-C^*-algebra is a (projective) limit of C^*-algebras in the category
of topological *-algebras. From the perspective of non-commutative
geometry, pro-C^*-algebras can be seen as non-commutative k-spaces. An
element of a pro-C^*-algebra is bounded if there is a uniform bound for
the norm of its images under any continuous *-homomorphism into a
C^*-algebra. The *-subalgebra consisting of the bounded elements turns
out to be a C^*-algebra. In this paper, we investigate pro-C^*-algebras
from a categorical point of view. We study the functor (-)_b that assigns
to a pro-C^*-algebra the C^*-algebra of its bounded elements, which is the
dual of the Stone-Cech-compactification. We show that (-)_b is a
coreflector, and it preserves exact sequences. A generalization of the
Gelfand duality for commutative unital pro-C^*-algebras is also presented.

Tuesday, March 21, 2006

Geoff Cruttwell, *A Generalisation of Normed Linear Spaces*

**Abstract**:
In 1973, Lawvere related metric spaces and enriched categories by showing
that a (quasi-)metric space is the same as a category enriched in
([0,\infty], \geq, +). I will be discussing a similar generalisation for
normed linear spaces, including some interesting examples and problems
with the theory.

Tuesday, March 28, 2006

Bob Paré, *Spans for Bicategories*

**Abstract**:
Spans for bicategories are just spans, but the 2-cells present features
which are perhaps a bit surprising. Isomorphic spans can look very
different, so does saying that spans are just spans make any sense?
This is joint work with Robert Dawson and Dorette Pronk.

Tuesday, April 4, 2006

Gillman Payette, *Notes from the Preservationist Underground: Level Compactness*

**Abstract**:
The concept of compactness is a necessary condition of any system
that is going to call itself a finitary method of proof. However,
it can also apply to predicates of sets of sentences in general
and in that manner it can be applied to a generalization of the
concept of a measure.

Tuesday, April 18, 2006

Benoit Valiron, *Quantum lambda calculus*

Tuesday, April 25, 2006

Peter Selinger, *Idempotents in dagger categories*

**Abstract**:
Dagger compact closed categories describe the main structure
of the category of finite dimensional Hilbert spaces. These categories
have been studied under a variety of names. In the 1980's,
mathematical physicists called them "*-categories" (a name derived
from C*-algebras); in the 1990's, John Baez called them "monoidal
categories with duals" (or for the less faint of heart: k-tuply
monoidal n-categories with duals); and most recently, Abramsky and
Coecke gave an interesting application to quantum protocols under the
name "strongly compact closed categories".
In this talk, I will define dagger compact closed categories and their
graphical language. I will show that the passage from "pure" to
"mixed" quantum computation can be described as a construction on
dagger compact closed categories called the CPM construction. I will
also discuss properties of idempotents in these categories, and the
viewpoint of "classical" data types as self-adjoint idempotents on
"quantum" types.

Tuesday, May 2, 2006

Jeff Eggar, *Open questions in category theory, relating to measure theory*

**Abstract**:
Vaughn Pratt recently asked on the category-theory
mailing-list what the big open problems in category theory are.
I'd like to weigh in with a few suggestions, and at the same time
set the stage for a future talk in which I will prove some theorems
about measure theory.

Tuesday, May 16, 2006

Francisco Marmolejo, *Active sums of groups*

**Abstract**:
The active sum of groups is a generalization of the direct sum of groups
that incorporates actions among the groups (we are basically thinking of
conjugation). In this talk we will explore some applications of homology
to this concept. We will report on some progress towards a conjecture
about metacyclic groups.

Tuesday, May 23, 2006

Jeff Egger, *Open questions in category theory, relating to measure theory:
Beyond the motivation*

Tuesday, May 30, 2006

Richard Wood, *Variation and Enrichment*

**Abstract**:
The parametrized 2-category constructions Fib/S, for S with finite limits,
and W-cat, for W a bicategory, are further unified by considering, for
fixed W, the 2-category of pseudo-functors H:A--->W which are locally discrete
fibrations. This 2-category is biequivalently described as the 2-category of
lax-functors W^co--->mat, where mat is the bicategory whose objects are sets
and whose hom-categories are given by mat(X,A)=set^{AxX}. (Both of these
2-categories require careful specification of their arrows and 2-cells, being
somewhat at odds with bicategorical orthodoxy.) The biequivalence is
a direct generalization of the Grothendieck biequivalence between fibrations
and CAT-valued pseudo-functors and is mediated by pulling back a universal
local discrete fibration mat_*--->mat. Further, the 2-category is also
biequivalent to the classical hat(W)-cat, where hat(W) is the bicategory whose
objects are those of W with hat(W)(w,x)=set^{W(w,x)^op}. We will show how
to recover the usual variable and enriched categories within this framework.
(Joint with JRB Cockett and SB Niefield)