Weak equivalence

In mathematics, a weak equivalence is a notion from homotopy theory which identifies complexes that have the same basic "shape" in terms of their homology groups. More precisely, if

is a morphism of chain complexes, then p is said to be a weak equivalence if and only if the induced maps

are isomorphisms. See also weak homotopy equivalence.

A fibration which is also a weak equivalence is also known as a trivial fibration. A cofibration which is also a weak equivalence is also known as a trivial cofibration.