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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-2-5,
author = {Takahisa Miyata},
title = {Fibrations in the Category of Absolute Neighborhood Retracts},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {55},
year = {2007},
pages = {145-154},
zbl = {1125.54011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-2-5}
}
Takahisa Miyata. Fibrations in the Category of Absolute Neighborhood Retracts. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 55 (2007) pp. 145-154. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba55-2-5/