Proto-exact and parabelian categories

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.