Varieties of simplicial groupoids. I: Crossed complexes. (English) Zbl 0888.55005
The paper is a contribution to the theory of algebraic models for homotopy types. W. G. Dwyer and D. M. Kan [Indag. Math. 46, 379-385 (1984; Zbl 0559.55023)] have shown that simplicial groupoids model all homotopy types completely. Partial models include crossed complexes which have been extensively studied by R. Brown and P. J. Higgins [J. Pure Appl. Algebra 47, 1-33 (1987; Zbl 0621.55009)]. In the paper, the simplicial groupoids that correspond to crossed complexes are shown to form a variety (epi-reflective subcategory) within the category of all simplicial groupoids and the corresponding verbal subgroupoid is identified.
Reviewer: K.H.Kamps (Hagen)
MSC:
| 55P15 | Classification of homotopy type |
| 55U35 | Abstract and axiomatic homotopy theory in algebraic topology |
| 18G30 | Simplicial sets; simplicial objects in a category (MSC2010) |
| 18G55 | Nonabelian homotopical algebra (MSC2010) |
Keywords:
simplicial groupoid; crossed complex; variety; verbal subgroupoid; homotopy type; groupoid \(T\)-complex; Dold-Kan theoremReferences:
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