On Cubical Sets of Quivers and Digraphs

TL;DR

This paper develops singular cubical homology theories for quivers and digraphs, introducing various realization notions and exploring their relationships, thereby extending algebraic topology concepts to combinatorial structures.

Contribution

It introduces multiple quiver realization concepts, defines new cubical homology theories for quivers and digraphs, and explores their interrelations and properties.

Findings

Defined several quiver realization notions

Established relations between different realizations

Introduced new path homology theories for cubical sets

Abstract

The singular cubical homology theory for the category of quivers or digraphs can be constructed similarly to the classical singular homology theory for topological spaces. The case of digraphs and quivers differs from the topological case due to the possibility of using a large number of non-isomorphic line digraphs that correspond to the unit interval in algebraic topology. In this paper we introduce several different notions of quiver realizations of a cubical set and we describe relations between them. We also define various singular cubical homology theories on quivers and digraphs. Moreover, using quiver realizations of cubical sets we define a collection of path homology theories on the category of cubical sets and we describe their properties.