Constructing linear bicategories

Abstract

Linearly distributive categories were introduced to model the tensor/par fragment of linear logic, without resorting to the use of negation. Linear bicategories are the bicategorical version of linearly distributive categories. Essentially, a linear bicategory has two forms of composition, each determining the structure of a bicategory, and the two compositions are related by a linear distribution. The main goal of this paper is to demonstrate that there are many examples of linear bicategories, which are obtained by considering quantales and quantaloids. It is standard in the field of monoidal topology that the category of quantale-valued relations is a bicategory. Here we begin by showing that a quantale is Girard if and only if the corresponding bicategory is a Girard quantaloid, which is an example of linear bicategory. The tropical and arctic semiring structures fit together into a Girard quantale, so this construction is likely to have multiple applications. More generally, we define LD-quantales, which are sup-lattices with two quantale structures related by a linear distribution, and their bicategorical analogue, linear quantaloids. We show that Q-Rel is a linear quantaloid if and only if Q is an LD-quantale. We then consider several standard constructions from enriched bicategory theory, and show that these lift to the linear quantaloid setting and produce new examples of linear bicategories. In particular, we consider linear Q-categories, matrices in Q and linear monads in Q, where Q is a linear quantaloid. We develop non-locally posetal examples as well, Quant, the bicategory of quantales, modules and module homomorphisms, and Qtld, the bicategory of quantaloids, modules and module homomorphisms. These turn out to be cyclic *-autonomous bicategories, which are in essence a closed version of linear bicategories.