Strong cofibrations and fibrations in enriched categories
It is well known that in the category of topological spaces, if \(A\rightarrow B\) and \(B\rightarrow Y\) are cofibrations and one of them is closed, then the induced morphism \(A\times Y\cup X\times B\rightarrow X\times Y\) is a cofibration. By introducing strong cofibrations, which are equivalent to closed cofibrations in the category of topological spaces, the authors give an analogous version of this useful ``push-out product theorem''. The results are proved in the setting of enriched categories which are complete and cocomplete, tensored and cotensored. Strong cofibrations are defined by a relative homotopy lifting property. The dual concept of a strong fibration allows the authors to prove a dual result. As a consequence, structures for doing homotopical algebra are obtained. Although in important cases Quillen structures [\textit{D. G. Quillen}, Homotopical algebra, Lect. Notes Math. 43 (1967; Zbl 0168.20903)] are not known to exist, it is possible to get fibration or cofibration structures in the sense of \textit{H.-J. Baues} [Algebraic Homotopy, Camb. Stud. Adv. Math. 15 (1989; Zbl 0688.55001)]. Finally, different examples are studied, in particular, the category of chain complexes with chain equivalences and the category of categories with equivalences of categories that are symmetric monoidal proper closed model categories.
scientific article; zbMATH DE number 1889275
scientific article