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TakuyaMurata (talk | contribs) →Abstract homotopy theory: mention simplicial set |
TakuyaMurata (talk | contribs) →Cofibration and fibration: based versions Tag: harv-error |
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A [[fibration]] in the sense of Serre is the dual notion of a cofibration: that is, a map <math>p : X \to B</math> is a fibration if given (1) a map <math>h_0 : Z \to X</math> and (2) a homotopy <math>g_t : Z \to B</math> such that <math>p \circ h_0 = g_0</math>, there exists a homotopy <math>h_t: Z \to X</math> that extends <math>h_0</math> and such that <math>p \circ h_t = g_t</math>. A basic example is a [[covering map]] (in fact, a fibration is a generalization of a covering map). If <math>E</math> is a [[principal bundle|principal ''G''-bundle]], that is, a space with a [[Group action#Remarkable properties of actions|free and transitive]] (topological) [[group action]] of a ([[topological group|topological]]) group, then the projection map <math>p: E \to X</math> is an example of a fibration.
There are also based versions of a cofibration and a fibration (namely, the maps are required to be based).<ref>{{harvnb|May|loc=Ch 8. § 3. and § 5.}}</ref>
=== Loop and suspension ===
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