Almost combinatorial accessible model categories

Question

Theorem: Assume VP. Let $\mathcal{M}$ be an accessible model category such that there exists a set of generating cofibrations $I$ and such that all objects are fibrant. Then it is combinatorial.

Proof: Consider the left determined model structure with respect to $I$: the minimal $I$-localizer is denoted by $\mathcal{W}_I$. Denote by $\mathcal{W}$ the class of weak equivalences of $\mathcal{M}$. Then $\mathrm{cof}(I) \cap \mathcal{W}_I \subset \mathrm{cof}(I) \cap \mathcal{W}$. Therefore all objects of the left determined model structure are fibrant. Since a model structure is characterized by its class of cofibrations and its class of fibrant objects, $\mathcal{M}$ is left determined, and therefore is combinatorial.

Question: Can we remove VP in the statement of the theorem ? I mean, is it as difficult as Smith's conjecture about left determined model structures ? Note that in the proof, I don't use the fact that the model category $\mathcal{M}$ is accessible, I only use the fact that the underlying category is locally presentable. The question is: is it easier when one already knows that $\mathcal{M}$ is accessible as a model category ?

Answer 1

Actually, you can, and you don't need accessibility (local presentability of the underlying category is enough).

Under your assumption, for each $i:A \to B$ a generating cofibration, take $B \coprod_A B \hookrightarrow I_A B \to B$ a cylinder object, and let $j_i : B \hookrightarrow I_A B$ be the first leg inclusion. Then the $j_i$ form a generating set of trivial cofibrations.

The proof essentially use that all objects are fibrant, and that you can consider the weak factorization system generated by the $j_i$ (so some smallness or local presentability assumption).

The key step in the proof is to observe that a map that has the RLP against the $j_i$ and is a weak equivalence is a trivial fibration. This easily follows from the fact that weak equivalences between fibrant objects have the "up to homotopy" lifting property against all cofibration and that lifting property against the $j_i$ is enough to rectify this in an actual lifting property. One then check by hand the cofibration, weak equivalence and the map with the lifting property against the $j_i$ forms a model structure, with the same cofirbation and weak equivalences as the one you started from, hence is the the same model structure.

I found it explicitly written out as corollary 3.2 of:

Valery Isaev, On fibrant objects in model categories, Theory and Applications of Categories, Vol. 33, 2018, No. 3, pp 43-66, journal page, arXiv:1312.4327