Mixing model structures (Q818346)

Let \(\mathcal C\) be a category which can be equipped with two Quillen model category structures: one with a class of weak equivalences \(W_1\) and a class of fibrations \(F_1\), and another with weak equivalences and fibrations given by classes of morphisms \(W_2\) and \(F_2\), respectively. The author shows that if we have inclusions \(W_1\subseteq W_2\) and \(F_1\subseteq F_2\) then there exists yet another ``mixed'' model category structure on \(\mathcal C\) with the class of weak equivalences \(W_2\) and the class of fibrations \(F_1\). An interesting application of this result is obtained by considering the category \({\mathcal T\!op}\) of topological spaces. By mixing the Strøm model category structure on \({\mathcal T\!op}\) (with homotopy equivalences and Hurewicz fibrations as, respectively, weak equivalences and fibrations) and the usual Quillen structure (with weak homotopy equivalences and Serre fibrations) one gets a model category where weak equivalences are weak homotopy equivalences, but fibrations are Hurewicz fibrations. Another paper by the author [Topology Appl. 153, No. 7, 1084--1099 (2006; Zbl 1097.55013)] can be used to produce several other examples of model category structures which can be mixed.