The weak categorical quiver minor theorem and its applications: matchings, multipaths, and magnitude cohomology

Luigi Caputi; Carlo Collari; Eric Ramos

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arXiv · 2401.01248

The weak categorical quiver minor theorem and its applications: matchings, multipaths, and magnitude cohomology

Luigi Caputi·Carlo Collari·Eric Ramos

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Abstract

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.

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BibTeX RIS

Luigi Caputi, Carlo Collari, Eric Ramos. 2024-01-02. The weak categorical quiver minor theorem and its applications: matchings, multipaths, and magnitude cohomology. https://arxiv.org/abs/2401.01248

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