Model Category - Homotopy and The Homotopy Category

Homotopy and The Homotopy Category

Given a model category, one can define an associated homotopy category by localizing with respect to the class of weak equivalences. Applying this to the category of topological spaces with the model structure given above, the resulting homotopy category is equivalent to the category of CW complexes and homotopy classes of continuous maps. This is also true for the model category of simplicial sets. In other words, the homotopy category of simplicial sets is also equivalent to the category of CW complexes and homotopy classes of continuous maps. Simplicial sets have nice combinatorial properties and are often used as models for topological spaces because of this equivalence of homotopy categories.

Note that the definition of homotopy category makes no mention of fibrations and cofibrations; this suggests that the information regarding homotopy is contained in the class of weak equivalences. Still, the classes of fibrations and cofibrations are useful in making constructions. Moreover, the "fundamental theorem of model categories" states that the homotopy category of C is always equivalent to the category whose objects are the objects of C which are both fibrant and cofibrant, and whose morphisms are left homotopy classes of maps (equivalently, right homotopy classes of maps) as defined above. (See for instance Model Categories by Hovey, Thm 1.2.10)