A semi-abelian approach to directed homology

TL;DR

This paper introduces a new homology theory for directed spaces using semi-abelian categories, which is invariant under directed homeomorphisms and computable via algebra quotients, with degenerate higher homology groups.

Contribution

It develops a novel directed homology theory based on semi-abelian algebra, providing invariance, computability, and insights into algebraic structures of directed spaces.

Findings

Directed homology HA is invariant under directed homeomorphisms.

HA_1 can be computed as a simple algebra quotient.

Higher homology groups HA_n (n≥2) are degenerate, as shown by an Eckmann-Hilton argument.

Abstract

We develop a homology theory for directed spaces, based on the semi-abelian category of (non-unital) associative algebras. The major ingredient is a simplicial algebra constructed from convolution algebras of certain trace categories of a directed space. We show that this directed homology HA is invariant under directed homeomorphisms, and is computable as a simple algebra quotient for $HA_1$. We also show that the algebra structure for $HA_n$, $n\geq 2$ is degenerate, through a Eckmann-Hilton argument. We hint at some relationships between this homology theory and natural homology, another homology theory designed for directed spaces. Finally we pave the way towards some interesting long exact sequences.