Fibrations in the Category of Absolute Neighborhood Retracts
Volume 55 / 2007
Abstract
The category $\mathsf{Top}$ of topological spaces and continuous maps has the structures of a fibration category and a cofibration category in the sense of Baues, where fibration = Hurewicz fibration, cofibration = the usual cofibration, and weak equivalence = homotopy equivalence. Concentrating on fibrations, we consider the problem: given a full subcategory $\mathcal C$ of $\mathsf{Top}$, is the fibration structure of $\mathsf{Top}$ restricted to $\mathcal C$ a fibration category? In this paper we take the special case where $\cal C$ is the full subcategory $\mathsf{ANR}$ of $\mathsf{Top}$ whose objects are absolute neighborhood retracts. The main result is that $\mathsf{ANR}$ has the structure of a fibration category if fibration = map having a property that is slightly stronger than the usual homotopy lifting property, and weak equivalence = homotopy equivalence.
