- E → B
satisfying the homotopy lifting property with respect to any space. Fiber bundles constitute important examples; but in homotopy theory any mapping is 'as good as' a fibration - i.e. any map can be decomposed as a homotopy equivalence into a "mapping path space" followed by a fibration.
A fibration with the homotopy lifting property for CW complexes (or equivalently, just cubes In) is called a Serre fibration, in honor of the part played by the concept in the thesis of Jean-Pierre Serre. This thesis firmly established in algebraic topology the use of spectral sequences, and clearly separated the notions of fiber bundles and fibrations from the notion of sheaf (both concepts together having been implicit in the pioneer treatment of Jean Leray). Because a sheaf (thought of as an étale space) can be considered a local homeomorphism, the notions were closely interlinked at the time.
The fibers are by definition the subspaces of E that are the inverse images of points b of B. Fibrations do not necessarily have the local cartesian product structure that defines the more restricted fiber bundle case, but something weaker that still allows 'sideways' movement from fiber to fiber. One of the main desirable properties of the Serre spectral sequence is to account for the action of the fundamental group of the base B on the homology of the total space E.
The projection map from a product space is very easily seen to be a fibration. Fiber bundles have 'local trivialization's — such cartesian product structures exist locally on B, and this is usually enough to show that a fiber bundle is a fibration. More precisely, if there are local trivializations over a "numerable open cover" of B , the bundle is a fibration. Any open cover of a paracompact space - for example any metric space, has a numerable refinement, so any bundle over such a space is a fibration. The local triviality also implies the existence of a well-defined fiber (up to homeomorphism), at least on each connected component of B.
In the Category of small categories, a functor p : E → C from a category E to a category C is a fibration iff for every object X of E and every map γ into pX in C there exists a cartesian morphism into X over γ (see also semidirect product).
Fibrations in closed model categories
A fibration in a closed model category C is an element of the class of morphisms of C called the fibrations of C. These are formally dual to the cofibrations in the opposite category Cop and in particular they are closed under composition and pullbacks. Any morphism in such a category can (by definition) be factored into the composition of a trivial cofibration followed by a fibration or a cofibration followed by a trivial fibration, where the word "trivial" indicates that the corresponding arrow is also a weak equivalence. The lifting property comes from one of the axioms for a model category which ties together fibrations and cofibrations by such lifts.