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High Quality Content by WIKIPEDIA articles! In mathematics, especially algebraic topology, a fibration is a surjective, continuous mapping p: E B satisfying the homotopy lifting property with respect to any space. Fiber bundles (over paracompact bases) constitute important examples. 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. (See homotopy fiber.) A surjective continuous mapping 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 etale space) can be considered a local homeomorphism, the notions were closely interlinked at the time.

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