# Almost combinatorial accessible model categories

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 ?

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

• I knew this paper but I had not paid attention to corollary 3.2. Thanks. – Philippe Gaucher Jul 4 at 13:21