In this chapter, continuing the construction of model structures for the category PC(M) of M-enriched precategories with variable set of objects, we define and discuss various classes of cofibrations. The corresponding classes of trivial cofibrations are then defined as the intersection of the cofibrations with the global weak equivalences defined in the preceding chapter. The fibrations are defined by the lifting property with respect to trivial cofibrations. It will have to be proven later that the class of trivial fibrations, defined as the intersection of the fibrations and the global weak equivalences, is also defined by the lifting property with respect to cofibrations.

In reality, we consider three model structures, called the injective, the projective, and the Reedy structures. These will be denoted by subscripts PCinj, PCproj and PCReedy respectively when necessary. They share the same class of weak equivalences, but the cofibrations are different. It turns out that the Reedy structure is the best one for the purposes of iterating the construction. That is a somewhat subtle point because the Reedy structure coincides with the injective structure when M lies in a wide range of model categories where the monomorphisms are cofibrations, see Proposition 13.7.2. When they are different it is better to take the Reedy route.

Skeleta and coskeleta

The definition of Reedy cofibrations, as well as the study of the projective ones, is based on consideration of the skeleta and coskeleta of objects in PC(M).