High Quality Content by WIKIPEDIA articles! In mathematics, a weak equivalence is a notion from homotopy theory which in some sense identifies objects that have the same basic "shape". This notion is formalized in the axiomatic definition of a closed model category. A closed model category by definition contains a class of morphisms called weak equivalences, and these morphisms become isomorphisms upon passing to the associated homotopy category. In particular, if the weak equivalences of two model categories containing the same objects and morphisms are defined in the same way, the resulting homotopy categories will be the same, regardless of the definitions of fibrations and cofibrations in the respective categories. Different model categories define weak equivalences differently. For example, in the category of (bounded) chain complexes, one might define a model structure where the weak equivalences are those morphisms.

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