The weak categorical quiver minor theorem and its applications: matchings, multipaths, and magnitude cohomology
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913183174754304 |
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| author | Caputi, Luigi Collari, Carlo Ramos, Eric |
| author_facet | Caputi, Luigi Collari, Carlo Ramos, Eric |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_01248 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The weak categorical quiver minor theorem and its applications: matchings, multipaths, and magnitude cohomology Caputi, Luigi Collari, Carlo Ramos, Eric Algebraic Topology Combinatorics Category Theory Representation Theory 18G85, 55F35 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. |
| title | The weak categorical quiver minor theorem and its applications: matchings, multipaths, and magnitude cohomology |
| topic | Algebraic Topology Combinatorics Category Theory Representation Theory 18G85, 55F35 |
| url | https://arxiv.org/abs/2401.01248 |