nLab proto-exact category

Idea

Proto-exact category is a nonadditive generalization of a Quillen exact category. The definition is self-dual as the notion of Quillen exact category is.

Definition

A proto-exact category is a pointed category $A$, with zero object $0$, together with two classes of morphisms, $I$ and $D$, called inflations and deflations such that

(i) each morphism $0\to U$ is an inflation and each morphism $U\to 0$ is a deflation

(ii) classes $I$ and $D$ are closed under composition and contain all isomorphisms

(iii) each square of the form

$$\array{ U &\rightarrow &V\\ \downarrow && \downarrow\\ W &\rightarrow & Z }$$

where the horizontal arrows are inflations and the vertical arrows are deflations is a pushout iff it is a pullback

(iv) Every diagram of the form $W\rightarrow Z\leftarrow V$ where $W\rightarrow Z$ is a deflation and $Z\leftarrow V$ a inflation may be completed to a biCartesian square of the form in (iii)

(v) Every diagram of the form $W\leftarrow U\rightarrow V$ where $W\leftarrow U$ is an inflation and $U\rightarrow V$ a deflation may be completed to a biCartesian square of the form in (iii)

Literature

The concept originates in

• Tobias Dyckerhoff, Mikhail Kapranov, Higher Segal spaces I, arxiv:1212.3563; now part of the book T. Dyckerhoff, M. Kapranov, Higher Segal spaces, Springer LNM 2244 (2019) doi

• Jaiung Jun, Matt Szczesny?, Jeffrey Tolliver, Proto-exact categories of matroids, Hall algebras, and K-theory Math. Z. 296 (2020) 147–167 doi
• Jaiung Jun, Matt Szczesny?, Jeffrey Tolliver, Proto-exact categories of modules over semirings and hyperrings, arXiv:2202.01573

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 \emph{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 $\mathcal{L}$ has a proto-exact structure, and furthermore that the subcategory of $\mathcal{L}$ consisting of finite lattices is equivalent to the category of finite $\mathcal{B}$-modules as proto-exact categories, where $\mathcal{B}$ is the Boolean semifield. We also discuss some relations between $\mathcal{L}$ 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 $\mathbf{K}$, a well-known relation between finite $\mathbf{K}$-modules and finite incidence geometries yields a combinatorial interpretation of exact sequences.

• Matthew B Young, Degenerate versions of Green’s theorem for Hall modules, J. Pure & Applied Algebra 225 (2021) 106557 doi

• J. Hekking, Segal objects in homotopical categories & K-theory of proto-exact categories, master thesis 2017 pdf