Hochschild homology, and a persistent approach via connectivity digraphs

Abstract

AbstractWe introduce apersistent Hochschild homologyframework for directed graphs. Hochschild homology groups of (path algebras of) directed graphs vanish in degree $$i\ge 2$$i≥2. To extend them to higher degrees, we introduce the notion ofconnectivity digraphs, and analyse two main examples; the first, arising from Atkin’sq-connectivity, and the second, here calledn-path digraphs, generalising the classical notion of line graph. Based on a categorical setting for persistent homology, we propose a stable pipeline for computing persistent Hochschild homology groups. This pipeline is also amenable to other homology theories; for this reason, we complement our work with a survey on homology theories of directed graphs.