A semi-abelian approach to directed homology

2023/01/16 by Éric Goubault, Goubault, Eric
Mathematics · Medicine · #Advanced Topics in Algebra #Algebraic Topology (math.AT) #FOS: Computer and information sciences #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Intracranial Aneurysms: Treatment and Complications #Logic in Computer Science (cs.LO)

paper · pdf · doi:10.48550/arxiv.2301.06409

openalex publication_date 2023/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 HA₁. We also show that the algebra structure for HA_n, n≥ 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.

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