2 - Basic Homotopical Algebra

Summary

The first half of this chapter is a recollection of homotopical algebra, after Quillen. We introduce weak factorisation systems and model category structures. We recall and prove the main results: explicit description of the Hom sets in the associated homotopy category, construction of derived functors, derived adjunction associated to a Quillen pair. We explain how to construct certain homotopy limits and colimits out of these notions. In particular, the notions of homotopy Cartesian square and of homotopy coCartesian square are discussed, as well as the notion of proper model category. The second half of the chapter gives a method to produce model structures on categories of presheaves from scratch. We observe that these methods do produce model category structures in families, giving rise to a refined description of the associated homotopy theory.