Hopf algebras for matroids over hyperfields

Recently, M.~Baker and N.~Bowler introduced the notion of matroids over hyperfields as a unifying theory of various generalizations of matroids. In this paper we generalize the notion of minors and direct sums from ordinary matroids to matroids over hyperfields. Using this we generalize the classical construction of matroid-minor Hopf algebras to the case of matroids over hyperfields

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

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 \begin{aligned} \pi ^s_n ({\mathbb {S}}) \hookrightarrow K_n (\mathbf {Mat}_{\bullet }) \end{aligned} 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.First author draf