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In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a model category. A model category is a category with classes of morphisms called weak equivalences, fibrations, and cofibrations, satisfying several axioms.
The Whitehead theorem states that a weak homotopy equivalence from one CW complex to another is a homotopy equivalence. (That is, the map f: X → Y has a homotopy inverse g: Y → X, which is not at all clear from the assumptions.) This implies the same conclusion for spaces X and Y that are homotopy equivalent to CW complexes.
A homotopy equivalence class of spaces is then called a homotopy type. There is a weaker notion: a map : is said to be a weak homotopy equivalence if : () is an isomorphism for each and each choice of a base point. A homotopy equivalence is a weak homotopy equivalence but the converse need not be true. Through the adjunction
For every space X one can construct a CW complex Z and a weak homotopy equivalence: that is called a CW approximation to X. CW approximation, being a weak homotopy equivalence, induces isomorphisms on homology and cohomology groups of X. Thus one often can use CW approximation to reduce a general statement to a simpler version that only ...
An equivalent definition is saying that a surjective map p: E → B is a quasifibration if the inclusion of the fibre p −1 (b) into the homotopy fibre F b of p over b is a weak equivalence for all b ∈ B. To see this, recall that F b is the fibre of q under b where q: E p → B is the usual path fibration construction. Thus, one has
Then the (true) homotopy category is defined by localizing the category of topological spaces with respect to the weak homotopy equivalences. That is, the objects are still the topological spaces, but an inverse morphism is added for each weak homotopy equivalence.
Left homotopy is defined with respect to cylinder objects and right homotopy is defined with respect to path space objects. These notions coincide when the domain is cofibrant and the codomain is fibrant. In that case, homotopy defines an equivalence relation on the hom sets in the model category giving rise to homotopy classes.
The homotopy category is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Two topological spaces X and Y are isomorphic in this category if and only if they are homotopy-equivalent.