Fibrations in the Category of Absolute Neighborhood Retracts

Takahisa Miyata¹

¹ Department of Mathematics and Informatics Graduate School of Human Development and Environment Kobe University Kobe, 657-8501, Japan

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 55 (2007) no. 2, pp. 145-154

Résumé

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.

Détail

DOI : 10.4064/ba55-2-5

Keywords: category mathsf top topological spaces continuous maps has structures fibration category cofibration category sense baues where fibration hurewicz fibration cofibration usual cofibration weak equivalence homotopy equivalence concentrating fibrations consider problem given full subcategory mathcal mathsf top fibration structure mathsf top restricted mathcal fibration category paper special where cal full subcategory mathsf anr mathsf top whose objects absolute neighborhood retracts main result mathsf anr has structure fibration category fibration map having property slightly stronger usual homotopy lifting property weak equivalence homotopy equivalence