In this chapter we review some of the higher categorical techniques which can be implemented, once we have the model structure. This covers what was done in my preprint on limits [235], but cast in the language of enrichment over a general M.

Fix a tractable left proper cartesian model category M. The terminology M-precategory will mean an object of PC(M) and an M-category will mean an M-precategory satisfying the Segal conditions. In other words, an M-category means a weakly M-enriched category. This notation is in effect throughout the remaining chapters.

When speaking of strictly M-enriched categories, that is to say precategories such that the Segal maps are isomorphisms, this will be specified explicitly.

The model structure on PC(M) is the Reedy model structure constructed in Theorem 19.2.1 unless otherwise specified.

Using the functor Set → M, which sends a set A to the colimit of * indexed by the discrete category A, a 1-category may be considered as an M-category.

In some places we will use a notation P for a cartesian left proper tractable model category. This is usually done in order to cast in general terms some discussions mainly intended for the case P = PC(M).

For the case of n-categories, a more iterative notation has been considered in Section 20.2, and this corresponds with the notation which was used in loc. cit. [235]. The reader may refer there for a version of the discussion in the old notation, but things seem to be clarified by considering a general enrichment.