Almost transferred model structures

Question

Let $F : \mathcal{C} \leftrightarrows \mathcal{D} : U$ be a Quillen adjunction between cofibrantly generated model categories. The model structure on $\mathcal{D}$ is called transferred if $U$ preserves and reflects weak equivalences and fibrations. It follows that cofibrations of $\mathcal{D}$ are generated by $F(\mathrm{I})$, where $\mathrm{I}$ is a set of generating cofibrations of $\mathcal{C}$.

Now, assume that cofibrations of $\mathcal{D}$ are generated by $F(\mathrm{I})$ and that $U$ reflects fibrant objects (or even all fibrations). Is the model structure on $\mathcal{D}$ necessarily transferred in this case? If these conditions hold and the transferred model structure exists, then it necessarily coincides with the given one. Thus, the question can be reformulated as follows. Is there a model category $\mathcal{D}$ satisfying conditions given above such that the transferred model structure on $\mathcal{D}$ does not exist?

Answer 1

This is too long for a comment, so I'm putting it here. I can think of three places to look for such an example. First, you could check out Example 2.8 in this paper I wrote with Michael Batanin: Bousfield Localization and Eilenberg-Moore Categories, arXiv:1606.01537.

In that example, we proved that the transferred model structure doesn't exist, so you could check if there is a different model structure on the category of operads valued in Ch($\mathbb{F}_2)$ that has the property you want.

Second, it is well-known that the transferred model structure on commutative differential graded algebras does not exist in characteristic $p>0$. So, CDGA($\mathbb{F}_p)$ doesn't have a transferred model structure. It does have a model structure discovered by Donald Stanley: Determining Closed Model Category Structures, (1998) (link), and you could check if Stanley's model structure has the property you want. Seems plausible.

Third, you could try to construct such an example using the techniques of Reid Barton at this answer: Counter-example to the existence of left Bousfield localization of combinatorial model category

You'd want to pick $C$ and $D$ to be very small, like Reid did. I would try to work this out, but I just don't have time (and probably won't all semester).