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.
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