Homotopy category

Summary

The homotopy category of a model category is obtained by making all weak equivalences invertible. It is the setting in which homotopy theory compares objects. Two objects are isomorphic in it when they are connected by a chain of weak equivalences. For topological spaces it can be described by CW complexes and homotopy classes of maps. Quillen's theorem identifies its morphisms with homotopy classes of maps between suitably replaced objects.