To prove that the classes of Reedy cofibrations and global weak equivalences give a model structure on PC(M), some computational work is needed. It turns out that the basic objects to be studied are the categories with an ordered set of objects x0, …, xn and morphisms other than the identity from xi to xj only when i < j. More precisely, we consider the free categories of this type obtained by specifying an object Bi ∈ M of morphisms from xi–1 to xi for 1 ≤ i ≤ n; then the object of morphisms from xi to xj should be the product of the Bk for k = i + 1, …, j. One of the main tasks is to look at a notion of precategory which corresponds to this notion of category, and to follow through explicitly the calculus of generators and relations.

In the next chapter will be our main calculation, which is what happens when one takes the product of two such categories. These considerations lead to a general theorem on products–the cartesian condition for PC(M). It turns out that the cartesian condition then implies the preservation of global weak equivalences under pushout which is the crucial hypothesis of Smith's recognition theorem.

Throughout, the notion of local weak equivalence on PC(X, M) is the one given by the model structures of Theorem 12.1.1 and 12.3.2.