Homotopy theory of cofibration categories
The author constructs a fibration category theory for the category of cofibration categories. Let \(\mathsf{CofCat}\) be the category of small cofibration categories, with morphisms given by exact functors between them. The definition of cofibration category structure adopted here is almost the dual version of Brown's original one [\textit{K. S. Brown}, Trans. Am. Math. Soc. 186, 419--458 (1973; Zbl 0245.55007)]. An exact functor of cofibration categories \(F:\mathcal{C}\rightarrow\mathcal{D}\) is defined to be a \textit{weak equivalence} if it induces an equivalence between the homotopy categories \(\mathrm{Ho}\mathcal{C}\rightarrow\mathrm{Ho}\mathcal{D}\). In order to define a \textit{fibration}, some properties are required from an exact functor. Those properties allow factorizations and the existence of pullbacks along cofibrations in \(\mathsf{CofCat}\). In addition, a technical property is also required in order to allow that acyclic fibrations (exact functors which are both a weak equivalence and a fibration) are stable under pullback. Then, the main theorem shows that the category \(\mathsf{CofCat}\) equipped with \textit{weak equivalences} and \textit{fibrations} has a complete fibration category structure.
scientific article; zbMATH DE number 6676577