Given our tractable, left proper and cartesian model category M, the main remaining problem in order to construct the global model structure on PC(M) is to consider the notion of interval, which should be an M-precategory (to be called Ξ(N|N′) in our notations below), weak equivalent to the usual category I with two isomorphic objects ν0, ν1 ∈ I, and with a single morphism between any pair of objects.

If A ∈ PC(M) is a weakly M-enriched category, an internal equivalence between x0, x1 ∈ Ob(A) is a “morphism from x0 to x1” (see (18.2.1) below), which projects to an isomorphism in the truncated category τ≤1(A). This terminology was introduced by Tamsamani [250]. It plays a vital role in the study of global weak equivalences. Essential surjectivity of a morphism f : A → B means (assuming that B is levelwise fibrant) that, for any object y ∈ Ob(B), there is an object x ∈ Ob(A) and an internal equivalence between f(x) and y.

Unfortunately, an internal equivalence between x0 and x1 in A doesn't necessarily translate into the existence of a morphism I → A. This will work after we have established the model structure on PC(M) if we assume that A is a fibrant object. However, in order to finish the construction of the model structure, we should start with the weaker hypothesis that A satisfies the Segal conditions and is levelwise fibrant.