  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.

Algebraic Topology;Logic in Computer Science