Friday, August 23, 2002
Abstract: The quadrality square combines (involutive) duality in linear structures such as the categories of abelian groups and sup-lattices with Girard's linear decomposition of cartesian closed categories and Stone duality for those categories. One example is the square

             ------------------------>     op
        Rel         equivalence         Rel

        ^  |                            |  ^
        |  |                            |  |
        |  | covariant        covariant |  |
 forget |-|| powerset          powerset |-|| forget
        |  |                            |  |
        |  |                            |  |
        |  V                            V  |
              contravariant powerset
              ------------------------>     op
         Set     monadic                 Set
              contravatiant powerset

in which the Stone duality at the bottom is Pare's theorem. Another, with order structures, is connected with the work of Marmolejo, Rosebrugh and Wood on completely distributive lattices, whilst there are others for locally compact topological spaces and for affine varieties. This talk will set out the details of some of these concrete examples, and attempt to extract from them axioms for a general notion of a category of "spaces". These axioms combine ideas of linear logic with monadic and comonadic adjunctions.

Tuesday, September 3, 2002
Sorin Dascalescu, Quantum Groups of Dimension 16 with the Chevalley Property

Tuesday, September 10, 2002
Claudia Centazzo, A duality relative to a limit doctrine
Abstract: We give a unified proof of Gabriel-Ulmer duality for locally finitely presentable categories, Adamek-Lawvere-Rosicky duality for varieties and Morita duality for presheaf categories.

Tuesday, September 17, 2002
Richard Wood, Wreaths
Abstract: For monads (or monoids or algebras) t (T) and s (S), it is classical that a distributive law \rho:ts ---> st (T\ten S ---> S\ten T) enables the construction of a monad (or monoid or algebra) structure on st (S\ten T) and that if certain compatibilities are demanded of the structure then all such arise from a distributive law. Yet, it seems that not all `structures on the tensor product' that arise in algebra from some sort of `twist' T\ten S ---> S\ten T satisfy such conditions. In a remarkable paper, `The formal theory of monads II', Lack and Street introduced the concept of a `wreath'. Its structure arrows and equations appear at first bizarre, so to enjoy the bizarre and avoid unnecessary suspense I will introduce them almost immediately in the talk and show how they give rise to a monad (or monoid or algebra) structure. I will hen show that the structure is not at all bizarre and has a certain completeness of purpose that was lacking for distributive laws.

Tuesday, October 1, 2002
Ross Street, Substitution, convolution & lax monoidal categories

Tuesday, October 8, 2002
Steve Lack, A Quillen Model Structure for 2-Categories

Tuesday, October 15, 2002
Peter Schoch, Adventures in non-classical logic part 4: 3-valued logic
Abstract: There is a great deal of nonsense talked about 3-valued logic. Two examples: (1)If an 'ordinary' sentence (i.e. a declarative) cannot be assigned either of the classical values it follows that it must receive some other, non-classical value. (2) 3-valued logic (and many-valued logic in general) is "really" two-valued because we talk about it in a 2-valued metalanguage. In this talk I attempt to transmute these two pig's ears into, if not a silk purse, then at least a useful back-pack. In the course of the talk I shall present the (only real) 3-valued logics due to {\L}ukasiewicz and Kleene and prove that they are the same, in a certain useful sense. I will also present a new kind of approach called "bi-semantics."

Tuesday, November 5, 2002
Barry Gardner, Rings Rich in Zero Dividing
Abstract: Bernhard Neumann has shown that every infinite subset of a group contains a pair of commuting elements if and only if the group is finite modulo its centre. Consideration of the wide ranging analogy between rings and groups based on the ring product and the group commutator suggestes the possibility that every infinite subset of a ring contains a pair of elements with zero product if and only if the ring is finite modulo its annihilator. In the special case of rings satisfying the identity x^2=0 this is indeed so and is proved by a relatively straightforward translation of the group theoty arguement. Does the condition on zero products force the identity? There is essentially one open case: that of an infinite ring with finite prime power exponent (and such a ring must be nil). Otherwise the answer is "yes".

Tuesday, November 12, 2002
Bob Rosebrugh, Partial information and the sketch data model
Abstract: Incomplete information is common in real-world databases, yet most data models do not intrinsically support references to missing data (often termed "null"s). The data model is analyzed assuming complete information, and support for nulls is retrofitted. The category-theoretic sketch data model is, instead, general enough to support references to missing information within itself. This talk will explore three approaches to incomplete information in the sketch data model. The approaches are ifferent, but sufficiently related that (under fairly strong hypotheses) they are Morita equivalent. The query languages arising are subtly different, and we explore some of these differences. (This is joint work with Michael Johnson.)

Tuesday, November 19, 2002
Dorette Pronk, Homology and Fractal Dimension
Abstract: Many fractals in 2-space have infinitely generated homology groups, so at first these groups may not seem very relevant. However, if one considers epsilon-neighborhoods of fractals, the smaller holes would be closed, so that one does obtain finite Betti numbers. However, such neighborhoods may also contain holes that were not in the original fractal, because the neighborhood may cover a small distance between two parts of the fractal. However, this would become clear once one would make epsilon smaller. This is the motivation for the definition of so called persistent betti numbers for fractals.

In this talk I will give an introduction to this concept, calculate some persistent betti numbers and show how they are related to the fractal dimension.

Tuesday, November 26, 2002
Richard Wood, Factorizing Regularity
Abstract:Carboni's KZ-doctrine ~R provides the free regular completion of a category with finite limits. In fact, with evident nomenclature, \reg = \lex^{~R}. In a recent paper we studied the possibility of extending Carboni's doctrine to \cat. The point is this: \lex = cat^E for a coKZ-doctrine E on \cat. Thus if there were a KZ-doctrine R on \cat whose restriction to \lex is ~R then there would be a distributive law ER--->RE over \cat giving \reg = \cat^{RE} and the concept of regularity would be factored in this precise sense.

In the previous paper we succeeded in extending ~R to a doctrine R on the 2-category \K consisting of all categories, functors that preserve kernel inclusions, and all natural transformations between these. Moreover, we identified \K^R as categories with regular factorizations in which regular epimorphisms are closed with respect to composition.

We are in the process of showing that there is a coKZ-doctrine L on \K for which \lex = \K^L. With this at hand we will have a distributive law LR--->RL over \K giving \reg = \K^{RL} and completing the main intent of the project.

(Joint work in progress with Claudia Centazzo)

Tuesday, January 7, 2003
Bob Paré, Some Properties of the Bicategory of Spans
Abstract: We shall study two different universal properties of the bicategory of spans. The second will lead to the notion of oplax bicategory. The free bicategory generated by an oplax one will be revealed for all to see.

Tuesday, January 14, 2003
Richard Wood, Modules between morphisms of bicategories
Abstract: We will define modules M:F--->G:\W--->\M, for (lax) morphisms of bicategories F and G so that in the special case where \M=\Sigma(ab), of the special case where \M has one object, of the special case where \W=1, a `module' is just the usual notion of F-G bimodule for rings F and G.

In the case of rings F,G, and H and bimodules M:F-->G and N:G--->H, the `tensor product' NM:F--->H is well known. We will explain this tensor product, in the opening generality, as further evidence in support of Paré's point of view on `multi-structures'

(This is joint work with J. Robin B. Cockett.)

Tuesday, January 28, 2003
Dorette Pronk, Conformal Field Theories as Nuclear Functors
Abstract: In this talk I will review the definitions of a tensored *-category and a nuclear ideal and show that Segal's "category" Cobord of disjoint unions of circles and diffeomorphism classes of surfaces between them is really a nuclear ideal in a larger category. (Segal's category does not have identity arrows. However, the correct larger category adds more than just identities.) One way to view conformal field theories has been as representations of Segal's "category", i.e. functors. I will show that one really wants nuclear representations of the larger category.

Tuesday, February 4, 2003
Bob Paré, A family of 2-categories in which every arrow has left and right adjoints

Tuesday, February 11, 2003
Kieth Johnson, Abel formal group laws and their role in algebraic topology

Tuesday, February 25, 2003
Peter Schoch, Everything you always wanted to know about modal logic but were afraid I'd tell you
Abstract: From the outside, modal logic has sometimes been compared to group representation theory---a highly technical area of interest to many potentially, but only a few specialists in actuality. In this talk I shall try to regain the interest of the mathematical/philosophical mass audience by concentrating on the issue "What is the logic of \emph{logical} necessity?" Along the way I shall show that all the usual answers are entirely wrong as are some of Dana Scott's answers. This would seem to be a consequence of previous authors failing to determine how such a question should be answered: namely that it should flow both from the fact that logical necessity is a kind of necessity and also that it is a kind of truth.

Tuesday, March 25, 2003
Janez Bernik, On semigroups of matrices with eigenvalue one in small dimensions
Abstract: In their recent paper "Irreducible semigroups of matrices with eigenvalue one" the authors J. Bernik, R. Drnovsek, T. Kosir and H. Radjavi, have shown, among other results, that for n=3 and n>4 there exist irreducible semigroups (and indeed groups) of n by n complex matrices with this spectral property. While it is easy to see that for n=2 no such semigroup exists, it is more difficult to see that this is also the case when n=4. The proof requires detailed analysis of the n=3 case combined with some results from the representation theory of finite groups and linear algebraic groups.

Tuesday, April 1, 2003
David Lever, Rings And Logical Morphisms
Abstract: Although the inclusion into a commutative ring of its ring of idempotents is not a ring homomorphism, it does preserve the elementary logic operations. This suggests an extension of the category of commutative rings to a category of commutative rings and logical morphisms. One is naturally led to examine the extension's dual to see if it is geometric. It is. I will prove this and discuss some of the reasons for studying the category of commutative rings and logical morphisms.