Having exhausted the theory of derived functors, we shift our focus to model categories, the context in which they are most commonly constructed. This contextualization appears in Section 11.3, where the definition of a model category finally appears, but first we focus on the parts of a model structure invisible to the underlying homotopical category. We think this perspective nicely complements standard presentations of model category theory, for example, [24, 36, 38]. (The newer [58] bears a familial resemblance to our presentation.) Those sources allow our treatment here to be quite brief. Where we have nothing of substance to contribute, rather than retrace well-trodden ground, we leave the standard parts of the theory to existing literature.

The reward for our work comes in the last section, in which we prove a theorem with a number of important consequences. Specifically, we describe the homotopical properties of the weighted limit and weighted colimit bifunctors in model category language. Using this, we give a simple proof of the homotopy finality theorem 8.5.6 and finally show that the different “op” conventions for the homotopy limit and homotopy colimit functors discussed in Remark 7.8.3 give weakly equivalent results. Finally, we prove that several familiar constructions for particular homotopy colimits have the appropriate universal properties.

Lifting problems and lifting properties

Classically, cofibrations and fibrations, technical terms in the context of any model structure, refer to classes of continuous functions of topological spaces characterized by certain lifting properties.