Proto-exact categories of matroids, Hall algebras, and \(K\)-theory. (English) Zbl 1442.18017
Summary: 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 T. Dyckerhoff and M. Kapranov [Higher Segal spaces. Cham: Springer (2019; Zbl 1459.18001)]. 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.
MSC:
| 18D99 | Categorical structures |
| 05B35 | Combinatorial aspects of matroids and geometric lattices |
| 16T30 | Connections of Hopf algebras with combinatorics |
| 19A99 | Grothendieck groups and \(K_0\) |
| 19D99 | Higher algebraic \(K\)-theory |
Keywords:
matroid; matroid strong maps; matroid-minor Hopf algebra; Hall algebra; proto-exact category; \(K\)-theoryCitations:
Zbl 1459.18001References:
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