Strong cofibrations and fibrations in enriched categories

Abstract

We define strong cofibrations and fibrations in suitably enriched categories using the relative homotopy extension resp. lifting property. We prove a general pairing result, which for topological spaces specializes to the well-known pushout-product theorem for cofibrations. Strong cofibrations and fibrations give rise to cofibration and fibration categories in the sense of homotopical algebra. We discuss various examples; in particular, we deduce that the category of chain complexes with chain equivalences and the category of categories with equivalences are symmetric monoidal proper closed model categories.