Weak homotopy equivalence

Summary

A weak homotopy equivalence is a continuous map that induces a bijection on path components and isomorphisms on all homotopy groups, at every base point. Every homotopy equivalence is a weak homotopy equivalence, and between CW complexes the converse holds by the Whitehead theorem. Every space is weakly equivalent to a CW complex, so weak equivalences are the natural notion of sameness in homotopy theory. They are the model for weak equivalences in abstract homotopy theory, such as in model categories.