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TakuyaMurata (talk | contribs) →CW complex: mention CW approximation Tag: harv-error |
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Another important result is the approximation theorem. First, the [[homotopy category]] of spaces is the category where an object is a space but a morphism is the homotopy class of a map. Then
{{math_theorem|name=[[CW approximation]]|math_statement=<ref>{{harvnb|May|loc=Ch. 10., § 5}}</ref> There exist a functor (called the CW approximation functor)
:<math>\Theta : \operatorname{Ho}(\textrm{spaces}) \to \operatorname{Ho}(\textrm{CW})</math>
from the homotopy category of spaces to the homotopy category of CW complexes as well as a natural transformation
:<math>\theta : i \circ \Theta \to \operatorname{Id},</math>
where <math>i : \operatorname{Ho}(\textrm{CW}) \hookrightarrow \operatorname{Ho}(\textrm{spaces})</math>, such that each <math>\theta_X : i(\Theta(X)) \to X</math> is a weak homotopy equivalence.
The above theorem justifies a common habit of working only with CW complexes. For example, given a space <math>X</math>, one can just define the homology of <math>X</math> to the homology of the CW approximation of <math>X</math>.
=== Cofibration and fibration ===
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