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Fibrations in the category of absolute neighborhood retracts

Miyata T.

Bulletin of the Polish Academy of Sciences. Mathematics

2007

Vol. 55, no 2

145-154

The category 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 C of Top, is the fibration structure of Top restricted to C a fibration category? In this paper we take the special case where C is the full subcategory ANR of Top whose objects are absolute neighborhood retracts. The main result is that 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.