• Corpus ID: 252212278

Constructing linear bicategories

@inproceedings{Blute2022ConstructingLB,
  title={Constructing linear bicategories},
  author={Richard Blute and Rose Kudzman-Blais and Susan B. Niefield},
  year={2022}
}

• R. Blute, Rose Kudzman-Blais, S. Niefield
• Published 13 September 2022
• Mathematics

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. After the initial paper on the subject, there was little further work as there seemed to be a lack of examples. The main goal of this paper is to demonstrate that there are in fact a great many examples, which are mainly obtained by extending familiar constructions from the (ordinary) bicategorical setting. While it is standard in the field of monoidal topology that the category of quantale-valued relations is a bicategory, we begin by showing that if the quantale is a Girard quantale, we obtain a linear bicategory. We further show that Q - Rel for Q a unital quantale is a Girard quantaloid if and only if Q is a Girard quantale. 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 show that Q - Rel is a linear bicategory if Q is an LD -quantale. We then consider several standard constructions from bicategory theory, and show that these lift to the linear bicategory setting and produce new examples of linear bicategories. In particular, we consider quantaloids. We first define the notion of a linear quantaloid Q and