For a finite quiver Q, we study the reachability category ReachQ. We in-vestigate the properties of ReachQ from both a categorical and a topological viewpoint. In particular, we compare ReachQ with PathQ, the category freely generated by Q. As a first application, we study the category algebra of ReachQ, 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 reacha-bility posets are isomorphic. As a second application, we define persistent Hochschild homology of quivers via reachability categories.