Tuesday, September 6, 2005
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
d


is 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

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)