2018/05/20 by Guram Donadze, Tim van der Linden, Tim Van der Linden
Mathematics · #Abelian group #Algebraic structures and combinatorial models #Cellular homology #Commutative Algebra and Its Applications #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Interpretation (philosophy) #Number theory #Singular homology #Spectral sequence #math.CT #math.KT #msc:18C15 #msc:18G10 #msc:18G40 #msc:20J06 #msc:55N35
paper · pdf · doi:10.1007/s40062-018-0225-3
published as J. Homotopy Relat. Struct. 14 (2019), 625--646 · 16 pages
arxiv created 2018/05/20 · openalex created_date 2018/06/01 · openalex publication_date 2018/11/27 · arxiv updated 2019/08/14 · openalex updated_date 2026/08/05
We introduce and study a homology theory of crossed modules with coefficients in an abelian crossed module. We discuss the basic properties of these new homology groups and give some applications. We then restrict our attention to the case of integral coefficients. In this case we regain the homology of crossed modules originally defined by Baues and further developed by Ellis. We show that it is an instance of Barr-Beck comonadic homology, so that we may use a result of Everaert and Gran to obtain Hopf formulae in all dimensions.