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

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.