A semi-abelian approach to directed homology

We develop a homology theory for directed spaces, based on the semi-abelian category of (non-unital) associative algebras, which we show encodes essentially natural homology (Dubut et al. in Appl Categ Struct 25(5):775–807, 2017. https://doi.org/10.1007/s10485-016-9438-y) and some of the composition pairing that refines it (Calk et al. in Homol Homotopy Appl, 2021). 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 HA₁. We also show that the algebra structure for HA_n, n≥2 is degenerate, through an Eckmann–Hilton argument. Finally we pave the way towards some interesting long exact sequences.