This paper examines the category Mat • of pointed matroids and strong maps from the point of view of Hall algebras. We show that Mat • 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 * (Mat • ) of Mat • via the Waldhausen construction, and show that it is non-trivial, by exhibiting injections π s n (S) ֒→ K n (Mat • ) from the stable homotopy groups of spheres for all n. Finally, we show that the Hall algebra of Mat • is a Hopf algebra dual to Schmitt's matroidminor Hopf algebra.
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