The theory of model categories is most effective when used in conjunction with the small object argument. In a nutshell, this says that by adding in enough copies of pushouts along arrows from a fixed set, we can ensure the right lifting property with respect to those arrows. This kind of discussion usually makes essential use of arguments about cardinality, and the most convenient categorical setting in which to do that is the notion of locally presentable category.

We therefore start with a review of that part of category theory. Our discussion is based in large part on the book of Adamek and Rosický [2] about locally presentable and accessible categories. Refer there for historical remarks about these notions. The applicability of this theory, in its abstract form, to model categories came out with J. Smith's notion of combinatorial model category [239] (see Beke [33], Dugger [95] and Rosický [223]), slightly modified by Barwick with his notion of tractable model category [18]. In turn, these authors were formalizing arguments which, for the basic cases derived from the category of simplicial sets, were due to Quillen [215], Bousfield and Kan [58], Jardine [153], Hirschhorn [144] and others.

The idea of “adding on many copies of something” translates into the notion of cell complex, which has long played an important role in algebraic topology.