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Normalisation for the fundamental crossed complex of a simplicial set. (English) Zbl 1184.55007
Summary: Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the fundamental crossed complex of a simplicial set. The main result is a normalisation theorem for this fundamental crossed complex, analogous to the usual theorem for simplicial abelian groups, but more complicated to set up and prove, because of the complications of the HAL and of the notion of homotopies for crossed complexes. We start with some historical background, {and give a survey of the required basic facts on crossed complexes.}

MSC:
55U10 Simplicial sets and complexes in algebraic topology
18D10 Monoidal, symmetric monoidal and braided categories (MSC2010)
18G30 Simplicial sets; simplicial objects in a category (MSC2010)
18G50 Nonabelian homological algebra (category-theoretic aspects)
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
55N10 Singular homology and cohomology theory
55N25 Homology with local coefficients, equivariant cohomology
55U99 Applied homological algebra and category theory in algebraic topology
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