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

This paper examines the category ${\mathbf{M}\mathbf{a}\mathbf{t}}_{\bullet }$ of pointed matroids and strong maps from the point of view of Hall algebras. We show that ${\mathbf{M}\mathbf{a}\mathbf{t}}_{\bullet }$ 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 $K_{\ast }({\mathbf{M}\mathbf{a}\mathbf{t}}_{\bullet })$ of ${\mathbf{M}\mathbf{a}\mathbf{t}}_{\bullet }$ via the Waldhausen construction, and show that it is non-trivial, by exhibiting injections $${\pi }_n^s(\mathbb{S})\hookrightarrow K_n({\mathbf{M}\mathbf{a}\mathbf{t}}_{\bullet })$$ from the stable homotopy groups of spheres for all $n$. Finally, we show that the Hall algebra of ${\mathbf{M}\mathbf{a}\mathbf{t}}_{\bullet }$ is a Hopf algebra dual to Schmitt’s matroid-minor Hopf algebra.