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arXiv · 1805.02281

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

Christopher Eppolito·Jaiung Jun·Matt Szczesny
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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 $$π^s_n (\mathbb{S}) \hookrightarrow K_n (\mathbf{Mat}_{\bullet})$$ 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.

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Christopher Eppolito, Jaiung Jun, Matt Szczesny. 2018-05-06. Proto-exact categories of matroids, Hall algebras, and K-theory. https://arxiv.org/abs/1805.02281

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