Tensor products and homotopies for \(\omega\)-groupoids and crossed complexes
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.
category of crossed complexes over a groupoid
scientific article; zbMATH DE number 4007245
scientific article