Weak equivalence (homotopy theory)

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

$p: A \rightarrow B\,$

where

$p_n: H_n(A) \rightarrow H_n(B).\,$

are isomorphisms for all n ≥ 0. However, this is not the only possible choice of weak equivalences for this category: one could also define the class of weak equivalences to be those maps that are chain homotopy equivalences of complexes.

For another example, the category of CW complexes can be given the structure of a model category where the weak equivalences are the weak homotopy equivalences i.e. those morphisms X → Y that induce isomorphisms in homotopy groups

$\pi_n(X, x) \cong \pi_n(Y, y)\,$

for all choices of basepoints x ∈ X, y ∈ Y, and all n ≥ 0.

A fibration which is also a weak equivalence is also known as a trivial (or acyclic) fibration. A cofibration which is also a weak equivalence is also known as a trivial (or acyclic) cofibration.