Weak equivalence (homotopy theory)

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 . A model category is a with classes of morphisms called weak equivalences, fibrations, and cofibrations, satisfying several axioms. The associated of a model category has the same objects, but the morphisms are changed in order to make the weak equivalences into isomorphisms. It is a useful observation that the associated homotopy category depends only on the weak equivalences, not on the fibrations and cofibrations. Model categories were defined by Quillen as an axiomatization of homotopy theory that applies to topological spaces, but also to many other categories in algebra and geometry. The example that started the subject is the category of topological spaces with Serre fibrations as fibrations and weak homotopy equivalences as weak equivalences (the cofibrations for this model structure can be described as the retracts of relative cell complexes X ⊆ Y). By definition, a continuous mapping f: X → Y of spaces is called a weak homotopy equivalence if the induced function on sets of path components is bijective, and for every point x in X and every n ≥ 1, the induced homomorphism on homotopy groups is bijective. (For X and Y path-connected, the first condition is automatic, and it suffices to state the second condition for a single point x in X.) For simply connected topological spaces X and Y, a map f: X → Y is a weak homotopy equivalence if and only if the induced homomorphism f*: Hn(X,Z) → Hn(Y,Z) on singular homology groups is bijective for all n. Likewise, for simply connected spaces X and Y, a map f: X → Y is a weak homotopy equivalence if and only if the pullback homomorphism f*: Hn(Y,Z) → Hn(X,Z) on singular cohomology is bijective for all n. Example: Let X be the set of natural numbers {0, 1, 2, ...} and let Y be the set {0} ∪ {1, 1/2, 1/3, ...}, both with the subspace topology from the real line.