On reachability categories, persistence, and commuting algebras of quivers

TL;DR

This paper explores the categorical and topological properties of reachability categories of quivers, linking them to path categories, commuting algebras, and poset incidence algebras, and introduces persistent Hochschild homology.

Contribution

It introduces a categorical framework for the commuting algebra of quivers, relating it to reachability categories and poset incidence algebras, and defines persistent Hochschild homology for quivers.

Findings

The commuting algebra of a quiver is Morita equivalent to the incidence algebra of its reachability poset.

Morita equivalence of commuting algebras corresponds to isomorphism of reachability posets.

The paper establishes a categorical approach to persistent Hochschild homology of quivers.

Abstract

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.