Simplicial approach to path homology of quivers, marked categories, groups and algebras

TL;DR

This paper generalizes the path homology theory to a simplicial framework, encompassing various structures like categories, groups, and algebras, and introduces a new homology for quivers.

Contribution

It develops a unified simplicial approach to path homology, extending existing theories and introducing a novel homology for quivers.

Findings

Unified framework for path homology across multiple structures

Introduction of square-commutative homology of quivers

Comparison with existing homology theories

Abstract

We develop a generalisation of the path homology theory introduced by Grigor'yan, Lin, Muranov and Yau (GLMY-theory) in a general simplicial setting. The new theory includes as particular cases the GLMY-theory for path complexes and new homology theories: path homology of categories with a chosen set of morphisms (marked categories) groups with a chosen subset (marked groups) and path Hochschild homology of algebras with chosen vector subspaces (marked algebras). Using our general machinery, we also introduce a new homology theory for quivers that we call square-commutative homology of quivers and compare it with the theory developed by Grigor'yan, Muranov, Vershinin and Yau.