Mixing model structures

Abstract

We prove that if a category has two Quillen closed model structures ( W 1 , F 1 , C 1 ) and ( W 2 , F 2 , C 2 ) that satisfy the inclusions W 1 ⊆ W 2 and F 1 ⊆ F 2 , then there exists a “mixed model structure” ( W m , F m , C m ) for which W m = W 2 and F m = F 1 . This shows that there is a model structure for topological spaces (and other topological categories) for which W m is the class of weak equivalences and F m is the class of Hurewicz fibrations. The cofibrant spaces in this model structure are the spaces that have CW homotopy type.