Tensor products and homotopies for \(\omega\)-groupoids and crossed complexes
type: Item
instance of: scholarly article
title: Tensor products and homotopies for \(\omega\)-groupoids and crossed complexes
zbMATH Open document ID: 0621.55009
DOI: 10.1016/0022-4049(87)90099-5
author: Philip J. Higgins
author: Ronald Brown
published in: Journal of Pure and Applied Algebra
publication date: 1987-01-01T00:00:00Z
review text: In his study of homotopy groups, additional structures on relative homotopy groups were observed by \textit{J. H. C. Whitehead} [Bull. Am. Math. Soc. 55, 453-496 (1949; Zbl 0040.388)]. These structures are built up out of crossed modules and give rise to the concept of crossed complexes. The authors have shown previously that crossed complexes satisfy a Van Kampen theorem which generalizes the relative Hurewicz theorem. In this paper the authors show that the category of crossed complexes over a groupoid has a symmetrical monoidal closed structure. This is built up from morphisms of crossed complexes, chain homotopies between them and higher homotopies. This is exposed by using cubical sets, as opposed to simplicial sets. Then the tensor product is introduced.
Mathematics Subject Classification ID: 55Q05
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 18D10
zbMATH DE Number: 4007245
zbMATH Keywords: crossed modules
zbMATH Keywords: crossed complexes
zbMATH Keywords: category of crossed complexes over a groupoid
zbMATH Keywords: symmetrical monoidal closed structure
zbMATH Keywords: higher homotopies
zbMATH Keywords: cubical sets
zbMATH Keywords: tensor product
reviewed by: Daniel H. Gottlieb
MaRDI profile type: Publication
full work available at URL: 0022-4049(87)90099-5
OpenAlex ID: W2153455826
cites work: Q3223557
cites work: Q3339777
cites work: Coproducts of crossed P-modules: applications to second homotopy groups and to the homology of groups
cites work: On the Connection between the Second Relative Homotopy Groups of Some Related Spaces
cites work: On the algebra of cubes
cites work: Colimit theorems for relative homotopy groups
cites work: Q4741410
cites work: Crossed complexes and chain complexes with operators
cites work: The classifying space of a crossed complex
cites work: Q3692477
cites work: Q5592283
cites work: Aspects of topoi
cites work: Lokal präsentierbare Kategorien. (Locally presentable categories)
cites work: Formal category theory: Adjointness for 2-categories
cites work: Q4172200
cites work: Residual Properties of Infinite Soluble Groups
cites work: An interpretation of the cohomology groups \(H^ n(G,M)\)
cites work: Crossed \(n\)-fold extensions of groups and cohomology
cites work: ABSTRACT HOMOTOPY
cites work: Q3491793
cites work: Q5840064
cites work: Cohomology of groups relative to a variety
cites work: Combinatorial homotopy. II
cites work: Q3682648
cites work: The non-abelian tensor product of groups and related constructions
description: scientific article; zbMATH DE number 4007245