July 8, 1997
George Gabor, Probability as Logic

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.

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.

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.

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.

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.

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.

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.

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.

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.