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.