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.

Recently, M. Baker and N. Bowler introduced the notion of matroids over hyperfields as a unifying theory of various generalizations of matroids. In this paper we generalize the notion of minors and direct sums from ordinary matroids to matroids over hyperfields. Using this we generalize the classical construction of matroid-minor Hopf algebras to the case of matroids over hyperfields. Date: December 27, 2017. 2010 Mathematics Subject Classification. 05E99(primary), 16T05(secondary). Key words and phrases. matroid, hyperfield, matroid over hyperfields, Hopf algebra, minor, direct sum.In this sense B and C B carry the same information as C B and B C respectively; these are thus said to determine the same matroid on E "cryptomorphically."Example 2.2. The motivating examples of matroids (hinted at above) are given as follows:(1) Let V be a finite dimensional vector space and E ⊆ V a spanning set of vectors.The bases of V contained in E form the bases of a matroid on E, and the minimal dependent subsets of E form the circuits of a matroid on E. Furthermore, these are the same matroid.(2) Let Γ be a finite, undirected graph with edge set E (loops and parallel edges are allowed). The sets of edges of spanning forests in Γ form the bases of a matroid on E, and the sets of edges of cycles form the circuits of a matroid on E. Furthermore, these are the same matroid (called the graphic matroid of Γ).One can define the notion of isomorphisms of matroids as follows.Definition 2.3. Let M 1 (resp. M 2 ) be a matroid on E 1 (resp. E 2 ) defined by a set B 1 (resp. B 2 ) of bases. We say that M 1 is isomorphic to M 2 if there exists a bijection f :In this case, f is said to be an isomorphism.Example 2.4. Let Γ 1 and Γ 2 be finite graphs and M 1 and M 2 be the corresponding graphic matroids. Every graph isomorphism between Γ 1 and Γ 2 gives rise to a matroid isomorphism between M 1 and M 2 , but the converse need not hold.Recall that given any base B ∈ B(M ) and any element e ∈ E \ B, there is a unique circuit (fundamental circuit ) C B,e of e with respect to B such that C B,e ⊆ B ∪ {e}.One can construct new matroids from given matroids as follows:Definition 2.5 (Direct sum of matroids). Let M 1 and M 2 be matroids on E 1 and E 2 given by bases B 1 and B 2 respectively. The direct sum M 1 ⊕ M 2 is the matroid on E 1 ⊔ E 2 given by the bases B = {B 1 ⊔ B 2 | B i ∈ B i for i = 1, 2}.Remark 2.6. One can easily check that M 1 ⊕ M 2 is indeed a matroid on E 1 ⊔ E 2 .Definition 2.7 (Dual, Restriction, Deletion, and Contraction). Let M be a matroid on a finite set E M with the set B M of bases and the set C M of circuits. Let S be a subset of E M .(1) The dual M * of M is a matroid on E M given by bases