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