Fibrations in Closed Model Categories
Fibrations of topological spaces fit into a more general framework, the so-called closed model categories. In such categories, there are distinguished classes of morphisms, the so-called fibrations, cofibrations and weak equivalences. Certain axioms, such as stability of fibrations under composition and pullbacks, factorization of every morphism into the composition of an acyclic cofibration followed by a fibration or a cofibration followed by an acyclic fibration, where the word "acyclic" indicates that the corresponding arrow is also a weak equivalence, and other requirements are set up to allow the abstract treatment of homotopy theory. (In the original treatment, due to Daniel Quillen, the word "trivial" was used instead of "acyclic.")
It can be shown that the category of topological spaces is in fact a model category, where (abstract) fibrations are just the Serre fibrations introduced above and weak equivalences are weak homotopy equivalences.
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