In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape".
It is a useful observation that the associated homotopy category depends only on the weak equivalences, not on the fibrations and cofibrations.
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[1]).
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.
(It is equivalent to consider "cochain complexes" of objects of A, where the numbering is written as simply by defining Xi = X−i.)