A semi-abelian approach to directed homology (Q6589168)

The author develops a novel homology theory \(HA\) tailored for directed spaces. This theory is based on the semi-abelian category of non-unital associative algebras. The core component of this approach is a simplicial algebra derived from convolution algebras associated with specific trace categories of a directed space. The directed homology \(HA\) is shown to be invariant under directed homeomorphisms and can be computed as a simple algebra quotient for \(HA_1\). For higher dimensions (\(HA_n\) for \(n \geq 2\)), the algebra structure is proven to be degenerate through an Eckmann-Hilton argument. \N\NThis work also explores the connections between this new homology theory and natural homology, another theory designed for directed spaces. The ultimate goal is to establish practical, computable invariants for directed topology, potentially implementable using existing computer algebra tools. Additionally, the paper hints at the possibility of interesting long exact sequences and aims to relate directed homology to other theories, such as persistence, semi-abelian categories, and representation theory of algebras.