Primitive path homology

TL;DR

This paper introduces a primitive path homology theory for simple digraphs, extending existing path homology concepts and enabling the study of paths with specific tail or head vertices.

Contribution

It develops a new primitive path homology framework that generalizes path homology for simple digraphs and explores its properties and relationships.

Findings

Primitive path homology coincides with path homology on asymmetric digraphs.

The theory allows constructing homologies for paths with fixed tail or head vertices.

Relationships between primitive path homology and existing path homology are established.

Abstract

In this paper we introduce a primitive path homology theory on the category of simple digraphs. On the subcategory of asymmetric digraphs, this theory coincides with the path homology theory which was introduced by Grigor'yan, Lin, Muranov, and Yau, but these theories are different in general case. We study properties of the primitive path homology and describe relations between the primitive path homology and the path homology. Let $a,b$ two different vertices of a digraph. Our approach gives a possibility to construct primitive homology theories of paths which have a given tail vertex $a$ or (and) a given head vertex $b$. We study these theories and describe also relationships between them and the path homology theory.