ATCAT 1999-2000
September 7, 1999
Mitja Mastnak, On extensions of Hopf algebras
Abstract: Extensions are used in algebra to build new objects out
of a pair of simple structures, or also to get information about
the structure of complicated objects by "decomposing" them into
pairs of simpler objects.
The aim of this presentation is to give an outline of an extension
theory for Hopf algebras. Extensions are characterized by certain
types of exact sequences. Every extension is associated with a
"so-called" abelian matched pair of Hopf algebras, and isomorphic
extensions belong to the same matched pair. The set of isomorphism
classes of extensions belonging to the same abelian matched pair
carries a Baer-type abelian group structure, and is isomorphic to
to the second cohomology group of that matched pair. This isomorphism
makes it possible to represent equivalence classes of extensions
by bicross products of abelian matched pairs. A couple of examples
will illustrate the results.
September 14, 1999
Peter Schotch, Worlds and Times
Abstract: This is another in the interminable series
of talks in which I wrestle with
the whole idea of modal logic (the logic of possibility and necessity.)
Mathematicians tend to skip over the part where one discovers the 'best way
to introduce modality' in their insensate rush to get to the 'good stuff'.
Philosophers, on the other hand, always pessimistically worry that there
won't be any good stuff, and so they get hung up on the early stages.
In the Fourteenth Century, Duns Scotus suggested that the proper
analysis of modality required not just moments of time but also
"moments of nature". In making this suggestion, he broke with an
influential view first presented by Diodorus in the early
Hellenistic period, and might even be said to have been the
inventor of "possible worlds". In this essay we take Scotus'
suggestion seriously devising first a "double-index" logic and
then introducing the temporal order. Finally, using the temporal
order, we define a "modal order". This allows us to present
modal logic without the usual interpretive questions arising concerning
the relation called variously "accessibility",
"alternativeness", and, "relative possibility".
September 21, 1999
David Lever,
Algebraic Neural Networks
Abstract:
A functor Z0:Bool ---> Ring provides the setting for the development of
algebraic neural networks. These networks are constructed from
aggregrates of polynomial representations of predicate formulas. We will
state and prove a learning theorem which provides the means for the
creation of algebraic neural networks over Boolean algebras. Any first
order theory gives rise to an algebraic neural network whose roots have
logical content. Numerical and differential methods may be used in many
cases to make or deny inferences at critical points of the polynomial
function.
September 28, 1999
RJ Wood,
Distributive Laws
Abstract:
We will begin by reviewing Beck's classical notion of
distributive law r:UD--->DU, for monads D and U (as
well as giving a very brief account of monads themselves).
Beck showed that distributive laws are coextensive with
two other kinds of structures. We will extend this list
with a further three and explain their significance.
Finally, we will apply the notion of distributive law
in some special contexts. In one case we will relate them
to the matched pairs of groups that arose in the recent
work of Mastnak; in another to factorization systems.
October 5, 1999
Dale Garraway,
Q-sets
Abstract:
It is known that the category of unital commutative C*-algebras is
equivalent to the category of compact Hausdorff spaces. From this Mulvey
introduced quantales as a non-commutative topology, the canonical example
being the lattice of subspaces of a C*-algebra. Extending the idea of
H-sets for a Heyting algebra, Nawaz('85) and Borceux('89,'93) each gave a
definition for the category of Q-sets on a right quantale (an idempotent
quantale with the top element acting as a right identity). We will show
that these two definitions are equivalent. Moreover, for Q a right
quantale any `reasonable' definition of the category of Q-sets is
equivalent to the category of H-sets for the locale of two sided elements
in Q. In the specific case where Q is the lattice of right ideals of a
C*-algebra A, then the category of Q-sets is equivalent to the category of
sheaves on the Hull-Kernal topology on A.
October 12, 1999
RJ Wood,
Distributive Laws (Continued)
Abstract:
I will continue my talk of two weeks ago on Distributive Laws.
In particular, I will begin with the new formulations of the
concept that I advertised last time but did not finish in the
talk. As an application, we will see the condition for a monad
on a set-like category (or an ordered-set-like category) to
extend to the corresponding category of relations. We will also
use the new formulations to simplify the concept in case the
monads in question are idempotent or `KZ'.
October 19, 1999
Margaret Beattie,
Twisting of comodule algebras and Hopf Galois objects
Abstract:
For a Hopf algebra H and an H-comodule algebra A, the notion of a twisting
of A
was defined some years ago by Beattie, Chen and Zhang, following work by
Zhang for
the case of graded rings. Basically the idea is to change the
multiplication of
A via a map from H to End(A) so that the resulting object is also an
H-comodule
algebra with the same H-comodule structure. If the twisting is invertible
in the
convolution algebra Hom(H,End(A)), then there is an isomorphism of
categories betweenthe categories of relative Hopf modules for A and the twisting of A.
Thus invertible twistings seem to be very strong. Question: if the
categories
of relative Hopf modules are isomorphic , are the algebras related by a
twisting??
I'd like to explain the basics of twistings and then consider twistings of
Hopf
Galois objects which are not crossed products, ie which do not have a
normal basis.
In this case, even if A is noncommutative and H is not cocommutative, a set of
cocycles shows up. This is joint work with Blas Torrecillas.
October 26, 1999
Moneesha Mehta,
Introduction to Lambda Calculus
November 2, 1999
Dietmar Schumacher,
Factorization Systems and Cotensors
November 9, 1999
Dietmar Schumacher,
Factorization Systems and Cotensors (Continued)
November 16, 1999
Dale Garraway,
G-Sets and Q-Sets
Abstract:
This talk will explore the relationships between group actions on a set
and Q-sets on a quantale. By taking the power set of the elements of an
arbitrary group we obtain an involutive quantale. It turns out that
G-sets are a specific type of Q-sets for this quantale. We next
generalize the notion of G-Sets to `G-Relations' the category of which is
equivalent to the category of Q-Sets.
November 30, 1999
RJ Wood,
Distributive laws and factorizations
Abstract:
(Joint work in progress with R Rosebrugh) Earlier this term,
in a talk on distributive laws, I mentioned that distributive
laws in the bicategory of matrices of sets give rise to a
category with a factorization system. This talk will attempt
to make that claim precise but will not assume a familiarity
with the bicategory of matrices of sets.
January 13, 2000
Peter Apostoli (Toronto),
Abstract approximation spaces of type-free sets: introducing the
penumbral modalities
Abstract:
The talk begins with a brief overview of the rough sets model and then
presents the domain theory. If I have time, I'll then show how the square
of modal opposition organizes the Russell set and its counterparts.
I also talk about the Cocchiarella result regarding the failure of the
identity of indicernibles in Fregean extensions of second order logic and
then attempt to motivate the rough set model by these considerations. After
discussing the standard epistemic interpretation of rough sets, I'll
introduce CFG as an example of abstract rough sets.
January 18, 2000
Robert Dawson,
What is a free double category like?
January 25, 2000
Tomaz Kosir,
On Gorenstein-Artin algebras
January 25, 2000
Robert Dawson,
What is a free double category like? (continued)
February 1, 2000
RJ Wood,
Tutorial on profunctors
February 8, 2000
Dietmar Schumacher,
Factorizations
February 15, 2000
Peter Schotch,
Rethinking Classical Metalogic
Abstract:
I have entered that period of my life during which I think about writing
textbooks (such a period is often refered to as one's 'declining years').
In getting together some notes for a textbook on philosophical logic, I
began to wonder how well the introductory material 'works' (on a
pedagogical level and also on the level of elegance, 'richness' simplicity,
and all the other dimensions upon which we judge theories. I'm now thinking
that on all these (and several others) elementary metalogic is sucking mud.
I have some suggestions for a rewrite/rethink that I would like to parade
for your viewing pleasure.
February 22, 2000
Dietmar Schumacher,
Factorizations
Abstract:
The talk will continue earlier ones on this subject. In particular,
the conditions under which the cancellativity laws for epics and
monics hold will be investigated.
March 7, 2000
Bob Paré,
Applications of Monad Theory to Topos Theory
Abstract:
I will recall Beck's and Butler's theorems and give
some well known applications to topos theory.
March 14, 2000
Bob Paré,
Applications of Monad Theory to Topos Theory 2
Abstract:
I will apply Beck's and Butler's theorems to the construction
of new toposes and the factorization of geometric morphisms.
March 21, 2000
Mitja Mastnak,
Galois theory in Hopf algebras
March 28, 2000
David Lever,
Nural Networks Over Affine Schemes
Abstract:
Affine hyperplanes of idempotent elements of a ring are equivalent
to vector fields defined on the spectrum of the ring. These hyperplanes
separate disjoint subspaces of the ring's spectrum. The main class of
examples of such hyperplanes are first order constructions in Sets.
June 20, 2000
David Lever,
Algebraic Quantification For Automated Reasoning
Abstract
Joint work with
Peter Schotch has resulted in a purely algebraic treatment of first order logic.
The talk will center on examples of how solving a systems of equations
associated to a statement provides a proof.