Let $\mathcal{C}$ be a cofibrantly-generated model category. My impression is that the following two conditions are highly correlated:

- $\mathcal{C}$ is right proper.
- There is an explicitly-describable set of generating acyclic cofibrations for $\mathcal{C}$.

(Of course, "explicitly-describable" is vague, but let's at least stipulate that "all acyclic cofibrations between small objects" (the sort of description one gets from Jeff Smith's recognition theorem) is *not* an explicit description *per se*.)

For example, the Quillen model structure satisfies both (1) and (2) (witness the horn inclusions), while the Joyal model structure satisfies neither (1) nor (2). Taking a Reedy model structure or projectively-inducing a model structure along an adjunction -- operations that preserve property (2) -- also preserve property (1). In fact, I don't know a single example of a model category $\mathcal{C}$ satisfying (1) but not (2) or (2) but not (1)! This leads to a vague question:

**"Question" A:** Does $(1) \Leftrightarrow (2)$ hold in some sense?

Here's a more precise, and seemingly stronger, formulation that I haven't been able to rule out. In lieu of explicit generating acyclic cofibrations, one often works with what Simpson calls a **pseudo-generating set**: a set of morphisms $S$ such that

if $Y$ is fibrant(including the case where $Y$ is terminal), then $X \to Y$ is a fibration iff it has the right lifting property with respect to the morphisms of $S$.

Cisinski's theory (nicely generalized by Olschok) often makes it easy to get one's hands on a pseudo-generating set even when a generating set is hard to describe. For example, the set $\{\Lambda^k[n] \to \Delta[n]\}_{n \in \mathbb{N},0 < k < n} \cup \{\Delta[0] \to I\}$ (where $I$ is the walking isomorphism) is a pseudo-generating set, but not a generating set, for the Joyal model structure. And Cisinski theory easily shows that the horn inclusions form a pseudo-generating set for the Quillen model structure. But in order to see that they are actually a generating set, one needs a nice functor like $Ex^\infty$; and such a nice functor automatically entails that one's model category is right proper. Somehow I suspect that the horn inclusions can't be so special, and I'm led to consider the condition

- Every pseudo-generating set in $\mathcal{C}$ is an actual set of generating acyclic cofibrations.

and to ask

**Question B:** Does $(1) \Leftrightarrow (3)$ hold?

even though I don't even know whether (3) holds for any $\mathcal{C}$ (unless every object is fibrant)! So I might as well also ask:

**Question C:** Is there an example of a model category $\mathcal{C}$ where not every object is fibrant where (3) holds?