
> 
> **Theorem**: Assume VP. Let $\mathcal{M}$ be an [accessible model category](https://ncatlab.org/nlab/show/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 ?
> 
> 
> 



Almost combinatorial accessible model categories
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](http://www.tac.mta.ca/tac/volumes/33/3/33-03abs.html), arXiv:[1312.4327](https://arxiv.org/abs/1312.4327)
> 
> 
> 


