  Building upon previous works of Proudfoot and Ramos, and using the
categorical framework of Sam and Snowden, we extend the weak categorical minor
theorem from undirected graphs to quivers. As case of study, we investigate the
consequences on the homology of multipath complexes; eg. on its torsion.
Further, we prove a comparison result: we show that, when restricted to
directed graphs without oriented cycles, multipath complexes and matching
complexes yield functors which commute up to a blow-up operation on directed
graphs. We use this fact to compute the homotopy type of matching complexes for
a certain class of bipartite graphs also known as half-graphs or ladders. We
complement the work with a study of the (representation) category of cones, and
with analysing related consequences on magnitude cohomology of quivers.
