Paul Taylor, QUADRALITY
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.