∗-Autonomous categories of bimodules

Abstract

Linear logic, a logical system developed by Girard [13] is a logic of resources which has elicited much interest from theoretical computer scientists because of its numerous potential applications. (See [36] for a brief overview and also [38].) It has also drawn the interest of logicians and category theorists. The connection with category theory comes about from the fact that the notion of *-autonomous category, due to Barr [2], provides a categorical model for a significant portion of linear logic. (See also his more recent exposition [3], as well as the work of Seely [37] and Blute [6,7]). Thus, attempts to find mathematical models for various aspects of linear logic center around *-autonomous categories and there have been interesting recent developments concerning connections with the theory of Petri nets [24,25]. Also, the notion of weakly distributive categories [l l] has been developed to model aspects of linear logic. The partially ordered (complete) models are called Girard quantales and they were first extensively studied by Yetter [40] and then by Rosenthal [31] (also see Chapter 6 in [30]). There are several interesting non-commutative Girard quantales, such as Rel(X), the relations on a set X, and Ord(P), the order ideals on a preordered set P. Until now, there has not been much in the way of examples of non-symmetric *-autonomous categories (other than partially ordered ones) as potential models for non-commutative linear logic. Recently, Barr [4] developed a non-symmetric version of the Chu construction for *-autonomous categories [2], and Blute has obtained non-symmetric *-autonomous categories by considering quantum groups (quasitriangular Hopf algebras) [S]. In this paper, we describe a-general way of constructing a special class of nonsymmetric *-autonomous categories, which we call cyclic, from a given *-autonomous category 9, by using enriched category theory and the calculus of 5?-bimodules. If Y is an autonomous category, we make the observation, following Lawvere [23], that