2022/02/03 by Jaiung Jun, Jun, Jaiung, Matt Szczesny +3 · 1 citation
Computer Science · Mathematics · #05B35 #06B99 #16Y20 (secondary) #16Y60 #18D99 (primary) #Category Theory (math.CT) #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic, programming, and type systems #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.2202.01573
openalex publication_date 2022/02/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 proto-exact 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 \mathbbB-modules as proto-exact categories, where \mathbbB 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 \mathbbK, a well-known relation between finite \mathbbK-modules and finite incidence geometries yields a combinatorial interpretation of exact sequences.