Many homotopy categories are homotopy categories
A well known theorem of \textit{A. Strøm} [Arch. Math. 23, 435--441 (1972; Zbl 0261.18015)] states that the category of topological spaces admits a model category structure where weak equivalences are homotopy equivalences and fibrations are Hurewicz fibrations. The paper generalizes this result as follows. Let \(\mathcal A\) be a complete and cocomplete category which is enriched, tensored and cotensored over the category of compactly generated topological spaces. For objects \(X, Y\in\mathcal A\) one can define a homotopy of morphisms from \(X\) to \(Y\) as a map \(X\otimes I \rightarrow Y\) (where \(I\) is the unit interval). This in turn can be used to define the notions of a homotopy equivalence and Hurewicz fibration in \(\mathcal A\). The author shows that these classes of morphisms define a model category structure on \(\mathcal A\) provided that an additional hypothesis concerning behavior of certain colimits is satisfied. Beside the category of topological spaces this result applies to the categories of \(G\)-spaces, prespectra, spectra, \(S\)-modules and others.
scientific article; zbMATH DE number 5013525
scientific article