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On reachability categories, persistence, and commuting algebras of quivers
Theory and applications of categories, Tome 41 (2024), pp. 426-448
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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.

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Classification : 16B50, 16P10, 05E10, 18B35
Keywords: reachability category, commuting algebras, persistent Hochschild homology 
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     author = {Luigi Caputi and Henri Riihim\"aki},
     title = {On reachability categories, persistence, and commuting algebras of quivers},
     journal = {Theory and applications of categories},
     pages = {426--448},
     year = {2024},
     volume = {41},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2024_41_a11/}
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Luigi Caputi; Henri Riihimäki. On reachability categories, persistence, and commuting algebras of quivers. Theory and applications of categories, Tome 41 (2024), pp. 426-448. http://geodesic.mathdoc.fr/item/TAC_2024_41_a11/