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On reachability categories, persistence, and commuting algebras of quivers

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Luigi Caputi, Henri Riihimäki

For a finite quiver Q, we study the *reachability category* Reach_Q. We investigate the properties of Reach_Q from both a categorical and a topological viewpoint. In particular, we compare Reach_Q with Path_Q, the category freely generated by Q. As a first application, we study the category algebra of Reach_Q, which is isomorphic to the *commuting algebra* of Q. As a consequence, we recover, in a categorical framework, previous results obtained by Green and Schroll; we show that the commuting algebra of Q is Morita equivalent to the incidence algebra of a poset, the reachability poset. We further show that commuting algebras are Morita equivalent if and only if the reachability posets are isomorphic. As a second application, we define *persistent Hochschild homology of quivers* via reachability categories.

Keywords:
reachability category, commuting algebras, persistent Hochschild homology

2020 MSC:
16B50, 16P10, 05E10, 18B35

*Theory and Applications of Categories,*
Vol. 41, 2024,
No. 12, pp 426-448.

Published 2024-04-11.

http://www.tac.mta.ca/tac/volumes/41/12/41-12.pdf

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