George Gabor,

*Probability as Logic*

July 8, 1997

Cristina Pedicchio (Trieste),

*Regular and Exact Locally Presentable Categories II*

September 9, 1997

Dale Garraway,

*Semi-Quantaloids (M.Sc. presentation)*

ABSTRACT: A semi-quantaloid is a semicategory enriched in the
monoidal category SUP of complete lattices and supremum preserving
functions. This notion is a multi-object generalization of quantales
which in turn is a
non-commutative generalization of complete Heyting algebras.
We construct
Q-valued sets, for Q a semi-quantaloid, which turn out to be a
generalization of H-valued sets, for H a complete Heyting algebra.
Since
it has been shown that H-valued sets are sheaves on H it is hoped
that
Q-valued sets are a generalization of the notion of Grothendieck
topos,
perhaps providing another approach to non-commutative topology.

September 23, 1997

Richard Wood,

*Monoidal Categories I*

ABSTRACT: A series of lectures on monoidal categories and their
applications. At first, purely expository and at an
elementary level with no background in category theory theory
assumed. Then
we proceed to more active areas of research
such as braided categories, quantum groups, and linear logic.

September 30, 1997

Bob Paré,

*Monoidal Categories II*

October 7, 1997

Richard Wood,

*Monoidal Categories III*

14 October 14, 1997

Bob Paré,

*Monoidal Categories IV*

October 21, 1997

Richard Wood,

*Monoidal Categories V*

October 28, 1997

Bob Paré,

*Monoidal Categories VI*

November 4, 1997

Richard Wood,

*Monoidal Categories VII*

November 18, 1997

Susan Niefield,

*Part I: Monoidal functors, monoids, and bimodules - a universal property*

ABSTRACT: Monoidal functors ``preserves'' monoids and their bimodules.
This
relationship is captured via an adjunction between the categories
of
monoids in V and monoidal categories over V.

November 18, 1997

Susan Niefield,

*Part II: Monoidal functors, categories, and bimodules - two universal properties*

ABSTRACT: Monoidal functors also 'preserves' enriched categories
and their
bimodules. Two generalizations of the adjunction in Part I will
be
presented. In one case, the adjunction is directly applied with
V-categories viewed as monoids. In the second, an adjunction
is
established for monoidal bicategories, and then applied to the
bicategory
of V-categories and bimodules.

November 25, 1997

Bob Paré,

*Monoidal Categories VIII*

December 2, 1997

Richard Wood,

*Monoidal Categories IX*

December 2, 1997

Dale Garraway,

*Semi-Qualnaloids*

January 13, 1998

David Lever,

*Algebraic Theorem Proving and Learning*

ABSTRACT: Well formed formulas of propositional and predicate
calculus are interpreted as polynomials over the integers or
real
numbers. The interpretation process is easily implemented on
computers. A logical implication is shown to be provable if and
only if its polynomial interpretation is the constant polynomial
1.
The polynomial language is used to study intra-deductibility
of
functional-link neural networks.

January 20, 1998

Moneesha Mehta,

*Discussion of Coherence Spaces with Reference to Linear Logic*

January 27, 1998

Dietmar Schumacher,

*The Proof of the Coherence Theorem for Monoidal Categories According to Power and Gordon*

January 27, 1998

Tomaz Kosir,

*On groups generated by elements of fixed prime order*

ABSTRACT: We will give a characterization of groups that are generated
by
elements of fixed prime order p and discuss some examples including
examples of
matrix groups. In particular, we will show that the special linear
group SLn(F)
is generated by elements of order p for each prime p and that
each element of
SLn(F) is product of 4 elements of order p. These are results
of joint work
with L. Grunenfelder, M. Omladic, and H. Radjavi.

February 3, 1998

Richard Wood,

*Equipments*

ABSTRACT: This talk will be addressed primarily to students and
newcomers. An
*equipment*, like a monoidal category or an enriched category,
is a category together with extra structure. Here the extra
structure is axiomatized with the intention of providing the
category in question with further arrows that are to be thought
of
as `relations'. Thus, for example, the category **set** of
sets
together with familiar relations is an equipment but so too is
**set**
together with partial functions and **set** together with spans.
Another motivating example is **cat**, the (mere) category
of
categories, together with profunctors which specializes somewhat
to
**ord**, the category of ordered sets, together with ordered
ideals.

An important liberty of the axioms is that the `relations' are
not assumed to admit a composition. This is somewhat dictated
by
the naturally occurring examples of *morphisms of equipments*
that arise in change of base problems. A goal of the talk is to
explain what is meant by an adjoint to a morphism of equipments.

February 10, 1998

Richard Wood,

*Equipments (Continuation)*

March 17, 1998

David Benson (Washington),

*The Topos of Three Coloured Graphs*

March 24, 1998

Bob Rosebrugh,

*Spans, Transition Systems and Minimal Realization*

ABSTRACT: Recently Katis, Sabadini and Walters have proposed the
(discrete,
cartesian) bicategory of spans of directed graphs as a suitable
algebra
for concurrent computation. Some of this work will be described.
It led
to joint work (in progress) with Walters on minimal realization
of behaviours in this context. A UIAO appears.

April 7, 1998

Moneesha Mehta,

*Four Propositional Logics*

ABSTRACT: Since Girard's introduction in 1987 of linear logic,
there has been much
discussion about its uses, its variants, and even its fundamental
logical
nature. This talk aims to present linear logic and its variants
in the
familiar settings of classical and intuitionistic logic, lattices,
and
category theory. In doing so, it is hoped that the nature of
linear logic
will become evident. Specifically, this talk will present logical
formulations of classical and intuitionistic propositional logics,
followed by formulations for classical linear logic and for
non-commutative, intuitionistic multiplicative linear logic i.e.,
Lambek
calculus, introduced by Lambek in 1958 and recognized by Girard
(and
others) by around 1990 to be the above mentioned variant of his
logic.
The talk will then go on to present posets and categories as
models for
these four propositional logics, and an equivalence between the
generic
category of logics and the category of posets.