On reachability categories, persistence, and commuting algebras of quivers

For a finite quiver $Q$, we study the reachability category ${\mathbf{R}\mathbf{e}\mathbf{a}\mathbf{c}\mathbf{h}}_Q$. We investigate the properties of ${\mathbf{R}\mathbf{e}\mathbf{a}\mathbf{c}\mathbf{h}}_Q$ from both a categorical and a topological viewpoint. In particular, we compare ${\mathbf{R}\mathbf{e}\mathbf{a}\mathbf{c}\mathbf{h}}_Q$ with ${\mathbf{P}\mathbf{a}\mathbf{t}\mathbf{h}}_Q$, the category freely generated by $Q$. As a first application, we study the category algebra of ${\mathbf{R}\mathbf{e}\mathbf{a}\mathbf{c}\mathbf{h}}_Q$, 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.