Why is this condition necessary for the existence of a transferred simplicial model structure?

Question

In chapter II of Goerss and Jardine's text on simplicial homotopy theory, they give a general theorem, Theorem 6.8, for transfer of simplicial model structures across a simplicial adjunction. This theorem generalizes Theorem 4.1, which concerns transfer of the standard simplicial model structure on $\operatorname{sSet}$ to the category $sC$ for some bicomplete category $C$. Theorem 6.8 says that given an adjunction $F:C\leftrightarrows D:G$ where $C$ is a $\beta$-cofibrantly-generated simplicial model category (for some cardinal $\beta$), $D$ is an arbitrary simplicial category, and $F$ preserves the simplicial tensoring, the simplicial model structure transfers provided three conditions hold:

1. $G$ preserves $\beta$-colimits;

2. If $f\in Mor(D)$ is in the saturation of $F(I)$, where $I$ is the set of generating fibrations for $C$, then $Gf$ is a cofibration;

3. A cofibration having the left lifting property with respect to all fibrations is a weak equivalence.

I'm trying to figure out why condition 2 is necessary. Conditions 1 and 3 are used in the standard fashion to do the small object argument, but the proof G&J outline makes no explicit use of 2. I went back to Theorem 4.1, and I couldn't find any use of a similar condition there, either. Intuitively, it doesn't really make sense to me: why should we care about the images of cofibrations under $GF$?

Is this condition truly necessary? If so, where and how is it used?

Answer 1

Since Goerss and Jardine do not give a (full) proof of this theorem, it is unclear how exactly this condition was intended to be used.

However, this type of construction (where weak equivalences and fibrations are created by a right adjoint functor) is known as a transferred model structure, and the relevant results originate with Kan. See, for example, Theorem 11.3.2 in Hirschhorn's book.

Assuming the small object argument can be performed (always true for cofibrantly generated model structures on locally presentable categories), the transferred model structure exists if and only if the functor G (or the functors G_i if there is more than one) sends transfinite compositions of cobase changes of elements of F(g) to weak equivalences in C, where g runs over generating acyclic cofibrations of the original model category C.

And if G_i preserves transfinite compositions (as is the case here), then the criterion can be further simplified to requiring that G sends cobase changes of elements of F(g) to weak equivalences in C.

This condition is necessary and sufficient, so yields the most general theorem of this type, and is considerably easier to verify than the conditions of Theorem II.6.8. Kan's theorem also easily implies the applications of Theorem II.6.8 in Goerss and Jardine's book.

(Goerss and Jardine's book was published in 1999, and a lot of simplification happened in this area since then, e.g., the work of Smith on combinatorial model categories.)