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

  • Chris Eppolito
  • Jaiung Jun
  • Matt SzczesnyEmail author
Article

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.

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 

Notes

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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesBinghamton UniversityBinghamtonUSA
  2. 2.Department of MathematicsNew PaltzUSA
  3. 3.Department of Mathematics and StatisticsBoston UniversityBostonUSA

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