Post date: 06-ago-2015 23:08:01
Following Quillen, we introduce the notion of cylinder objects in a model category, and the associated (generalized) notion of homotopy. We show that fibrations in a model category behave like hurewicz fibrations, in that they satisfy the homotopy lifting property. We show that under mild hypothesis homotopy defines an equivalence relation and is independent of the chosen cylinder.
As a corollary we obtain a generalization of Whitehead theorem, between nice objects, that is fibrant-cofibrant objects, an arrow is a weak equivalence if and only if is a homotopy equivalence. Then we show that the homotopy category of a model category exists (i.e. is locally small), and it can be constructed by taking homs the homotopy equivalence classes of arrows between fibrant-cofibrant replacements.