In this chapter, we consider the cartesian product of two M-enriched precategories. On the one hand, we would like to maintain the cartesian or product condition (PROD) for the new model category we are constructing. On the other hand, compatibility with cartesian product provides the main technical tool we need in order to study pushouts by global weak equivalences in PC(M). For a fixed set of objects X, recall that we already have the projective (and sometimes injective) model structures of Theorem 12.1.1, as well as the Reedy structure PCReedy(X, M) of Theorem 12.3.2. We will mainly be using the Reedy structure.

The situation can be simpler to understand when M consists of presheaves over a connected category and the cofibrations are the monomorphisms; then Proposition 13.7.2 applies to say that the Reedy and injective structures are the same. That case would be sufficient for iterating our main construction PC, a point of view which allows the reader to replace Reedy cofibrations by injective ones the first time through.

In this chapter is the place where we really use the full structure of the category Δ, as well as the unitality condition. At the end of the chapter, we'll discuss some counterexamples showing why these aspects are necessary.

Products of sequentially free precategories

Suppose X = {x0, …, xm} and Y = {y0, …, yn} are finite linearly ordered sets, and (X, A) and (Y, B) are sequentially free M-precategories, with respect to the orderings.