Quiver matroids -- Matroid morphisms, quiver Grassmannians, their Euler characteristics and $\mathbb{F}_1$-points

In this paper, we introduce morphisms for matroids with coefficients (in the sense of Baker and Bowler) and quiver matroids. We investigate their basic properties, such as functoriality, duality, minors and cryptomorphic characterizations in terms of vectors, circuits and bases (a.k.a. Grassmann-Pl\"ucker functions). We generalize quiver matroids to quiver matroid bundles and construct their moduli space, which is an $\mathbb{F}_1$-analogue of a complex quiver Grassmannian. Eventually we introduce a suitable interpretation of $\mathbb{F}_1$-points for these moduli spaces, so that in 'nice' cases their number is equal to the Euler characteristic of the associated complex quiver Grassmannian.