Proto-exact categories of matroids, Hall algebras, and K-theory
Files
First author draft
Date
2019
Authors
Szczesny, Maciej
Jun, Jaiung
Eppolito, Chris
Version
First author draft
OA Version
Citation
Maciej Szczesny, Jaiung Jun, Chris Eppolito. 2019. "Proto-exact categories of matroids, Hall algebras, and K-theory." Mathematische Zeitschrift. https://10.1007%2Fs00209-019-02429-z
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
\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.