Homotopy Category - Generalizations

Generalizations

The above definition of the homotopy of topological spaces is a special case of the more general construction of the homotopy category of a model category. Roughly speaking, a model category is a category C with three distinguished types of morphisms called fibrations, cofibrations and weak equivalences. Localizing C with respect to the weak equivalences yields the homotopy category.

This construction, applied to the model category of topological spaces, gives back the homotopy category outlined above. Applied to the model category of chain complexes over some commutative ring R, for example, yields the derived category of R-modules. The homotopy category of chain complexes can also be interpreted along these lines.