Proto-exact categories of matroids, Hall algebras, and K-theory

TL;DR

This paper introduces a new categorical framework for pointed matroids, explores their algebraic K-theory, and links Hall algebras to classical structures like the stable homotopy groups of spheres.

Contribution

It establishes that the category of pointed matroids is a finitary proto-exact category, defines its algebraic K-theory, and relates its Hall algebra to known Hopf algebras and homotopy groups.

Findings

The category of pointed matroids is a finitary proto-exact category.

The K-theory of pointed matroids is non-trivial and contains the stable homotopy groups of spheres.

The Hall algebra of pointed matroids is a Hopf algebra dual to Schmitt's matroid-minor Hopf algebra.

Abstract

This paper examines the category $\mathbf{Mat}_{\bullet}$ of pointed matroids and strong maps from the point of view of Hall algebras. We show that $\mathbf{Mat}_{\bullet}$ has the structure of a finitary proto-exact category - a non-additive generalization of exact category due to Dyckerhoff-Kapranov. We define the algebraic K-theory $K_* (\mathbf{Mat}_{\bullet})$ of $\mathbf{Mat}_{\bullet}$ via the Waldhausen construction, and show that it is non-trivial, by exhibiting injections $$\pi^s_n (\mathbb{S}) \hookrightarrow K_n (\mathbf{Mat}_{\bullet})$$ from the stable homotopy groups of spheres for all $n$. Finally, we show that the Hall algebra of $\mathbf{Mat}_{\bullet}$ is a Hopf algebra dual to Schmitt's matroid-minor Hopf algebra.