Proto-exact and parabelian categories
Sergey Mozgovoy
TL;DR
Proto-exact and parabelian categories provide a non-additive framework mirroring exact and quasi-abelian categories, enabling algebraic $K$-theory and Hall algebra constructions via the Waldhausen $S$-construction and a $2$-Segal structure. The paper proves that every parabelian category carries a canonical proto-exact structure and develops a broad suite of examples, from pointed sets and normed/Euclidean spaces to Hermitian vector bundles and matroids, extending to finitary algebraic categories through monads and valuation monads. It establishes equivalence with existing proto-exact definitions and gives practical criteria to determine parabelianity in finitary algebraic settings, including convex and absolutely convex spaces. By connecting valuation monads and monadic constructions to Arakelov-type geometric contexts, the work broadens non-additive homological methods and paves the way for non-additive $K$-theory and Hall algebra frameworks.
Abstract
Proto-exact and parabelian categories serve as non-additive analogues of exact and quasi-abelian categories, respectively. They give rise to algebraic K-theory and Hall algebras similarly to the additive setting. We show that every parabelian category admits a canonical proto-exact structure and we study several classes of parabelian categories, including categories of normed and Euclidean vector spaces, pointed closure spaces and pointed matroids, Hermitian vector bundles over rings of integers. We also examine finitary algebraic categories arising in Arakelov geometry and provide a criterion for determining when such a category is parabelian. In particular, we prove that the categories of pointed convex spaces and absolutely convex spaces are parabelian.