Proto-exact categories of modules over semirings and hyperrings

Abstract

Proto-exact categories, introduced by Dyckerhoff and Kapranov, are a generalization of Quillen exact categories which provide a framework for defining algebraic K-theory and Hall algebras in a non-additive setting. This formalism is well-suited to the study of categories whose objects have strong combinatorial flavor.In this paper, we show that the categories of modules over semirings and hyperrings -algebraic structures which have gained prominence in tropical geometry -carry proto-exact structures.In the first part, we prove that the category of modules over a semiring is equipped with a protoexact structure; modules over an idempotent semiring have a strong connection to matroids. We also prove that the category of algebraic lattices L has a proto-exact structure, and furthermore that the subcategory of L consisting of finite lattices is equivalent to the category of finite B-modules as protoexact categories, where B is the Boolean semifield. We also discuss some relations between L and geometric lattices (simple matroids) from this perspective.In the second part, we prove that the category of modules over a hyperring has a proto-exact structure. In the case of finite modules over the Krasner hyperfield K, a well-known relation between finite K-modules and finite incidence geometries yields a combinatorial interpretation of exact sequences.