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

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.

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Notes

  1. 1.

    Not all matroids arise in this way. For more details, see [21].

  2. 2.

    This is exactly a loop of the graph in the graphic case, Example 2.1.2.

  3. 3.

    Not all strong maps between matroids arise in this way (even if the matroids themselves do).

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Acknowledgements

The first author would like to thank L. Anderson for helpful comments. The second author was supported by an AMS-Simons Travel grant, and the paper was written when the second author was working at Binghamton University. The third author is grateful to Tobias Dyckerhoff for explanations regarding the paper [7] and for the support of a Simons Foundation Collaboration Grant. The authors collectively thank an anonymous referee for many helpful suggestions, including the observation in Remark 6.5.

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Correspondence to Matt Szczesny.

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Eppolito, C., Jun, J. & Szczesny, M. Proto-exact categories of matroids, Hall algebras, and K-theory. Math. Z. 296, 147–167 (2020). https://doi.org/10.1007/s00209-019-02429-z

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Keywords

  • Matroid
  • Matroid strong maps
  • Matroid-minor Hopf algebra
  • Hall algebra
  • Proto-exact category
  • K-theory

Mathematics Subject Classification

  • Primary 18D99
  • Secondary 05B35
  • 16T30
  • 19A99
  • 19D99