Enriched categories and tropical mathematics

TL;DR

This survey explores the deep connections between enriched category theory over quantales and tropical mathematics, highlighting categorical constructions, reformulations of tropical objects, and the structure of semimodules as tropical vector spaces.

Contribution

It provides a comprehensive overview linking enriched categories and tropical mathematics, including new insights into semimodules and categorical reformulations of tropical structures.

Findings

Reformulation of tropical polytopes via category-theoretic constructions

Identification of complete semimodules with skeletal complete categories

Illustration of the ubiquity of categorical constructions in tropical mathematics

Abstract

This is a survey paper on the connection of enriched category theory over a quantale and tropical mathematics. Quantales or complete idempotent semirings, as well as matrices with coefficients in them, are fundamental objects in both fields. We first explain standard category-theoretic constructions on matrices, namely composition, right extension, right lifting and the Isbell hull. Along the way, we review known reformulations (due to Elliott and Willerton) of tropical polytopes, directed tight spans and the Legendre--Fenchel transform by means of these constructions, illustrating their ubiquity in tropical mathematics and related fields. We then consider complete semimodules over a quantale $\mathcal{Q}$, a tropical analogue of vector spaces over a field, and mention Stubbe's result identifying them with skeletal and complete $\mathcal{Q}$-categories. With the aim to bridge a gap between enriched category theory and tropical mathematics, we assume no knowledge in either field.