Fibration

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In algebraic topology, a fibration is a continuous mapping

YX

satisfying the homotopy lifting property. Fiber bundles constitute important examples; but in homotopy theory any mapping is 'as good as' a fibration.

A fibration with the homotopy lifting property for CW complexes is often called fibration in the sense of Serre, in honour of the part played by the concept in the thesis of Jean-Pierre Serre. This established in algebraic topology the notion of spectral sequence, in a way that separated it out from the notion of sheaf (both concepts together having been implicit in the pioneer treatment of Jean Leray). Because a sheaf is 'as good as' a local homeomorphism, the notions seemed closely interlinked at the time.

The geometry of a fibration is, by definition, to do with being able to lift up homotopies in the base X into the fibering space Y. The fibers are by definition the subspaces of Y that are the inverse images of points x of X. We are not in this case given a local cartesian product structure (which defines the more restricted fiber bundle case), but something possibly 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 X on the homology of the total space Y.

In order to pass from a fibration to a fiber bundle, one needs a local trivialization — in other words the existence of such a cartesian product structure, locally on X. This then implies the existence of a well-defined fiber (up to homeomorphism), at least on each connected component of X.

In the category theory, a functor p : EC 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.

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