Explicit generating acyclic cofibrations and right properness of a model category

Question

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

1. $\mathcal{C}$ is right proper.
2. 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

3. 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?

Answer 1

If $\mathcal C$ is the category of simplicial presheaves on a small category $A$ equipped with the injective model structure, it is proper, but it is very unlikely that we will get "explicit generating (trivial) cofibrations" without further assumptions (for instance on the indexing category $A$, such as being elegant in the sense of Rezk and Bergner). And it is hard to define the adjective "explicit" mathematically with the level of generality you are aiming at...

That said, here are classes of examples which gives a hint about the freedom we have, and in particular, about the fact that having explicit generators and right properness are not directly related.

First, the left Bousfield localizations of the Kan-Quillen model structure on the category of simplicial sets which are proper are precisely the nullifications. This gives quite a lot of freedom to mess around.

Second, there are proper model structures with explicit generators but no nice functors such as the $Ex^\infty$-functor: for instance cubical sets.

To come back to the general case, it is a good thing to remember what properness is about: right properness means that the formation of slices is compatible with weak equivalences (i.e. pulling back along any weak equivalence $X\to Y$ induces a right Quillen equivalence from $\mathcal C/Y$ to $\mathcal C/X$). In particular, the canonical model structures induced on slices $\mathcal C$ are rather relevant when it comes to express whether the model structure $\mathcal C$ is right proper or not.

If ever we have "explicit" pseudo-generators, in practice, we can define yet another model structure on each slice $\mathcal C/X$, which is fully characterized through the following: the cofibrations are the maps which are cofibrations in $\mathcal C$, while the fibrant objects are the maps $E\to X$ which have the right lifting property with respect to the pseudo-generating set of trivial cofibrations. The usual sliced model structure on $\mathcal C/X$ is thus a left Bousfield localization of the latter, but there is no reason that they agree. In fact, they agree for all $X$ if and only if the pseudo-generators actually are generators of trivial cofibrations.

Here is an enlightening example. Horn inclusions of the form $\Lambda^n_k\to\Delta^n$ for $n\geq 1$ and $0\geq k< n$ do form a pseudo-generating set for the usual Kan-Quillen model structure. Moreover, on slices, the model structure induced by these pseudo generators are fully documented: they are the "contravariant model structures" modelling presheaves on quasi-categories. In particular, they are not always proper and they coincide with the usual sliced model structures only when we slice over simplicial sets whose fundamental category (obtained through the left adjoint of the nerve) is a groupoid. In particular, the fact that these pseudo-generators fail to be generators is not an accident nor a failure: it is needed because we do not want all presheaves to be locally constants!

Answer 2

The answer to Question A is no. A counterexample is provided by left Bousfield localization. Let $M$ be left proper and cellular. Let $C$ be a set of maps in $M$. Hirschhorn's machine for left Bousfield localization $L_C(M)$ gives some set of generating trivial cofibrations, but it's hard to describe. If it's easy enough to describe that you think it's "explicitly describable," then this is your counterexample, because it's well-known that $L_C(M)$ can fail to be right proper (Hirschhorn has examples). If you think the generating trivial cofibrations are not "explicitly describable," I can still give you a counterexample, using Section 9 of Bousfield's paper "On the telescopic homotopy theory of spaces." In that paper, Bousfield gives conditions under which $L_C(M)$ must be right proper, but this does not appear to provide control over the generating trivial cofibrations: they are as crazy as in Hirschhorn's case.

Moving on to question B, evidence against it is provided by the theory of right Bousfield localizations $R_K(M)$. These are always right proper, but need not be cofibrantly generated. The generating trivial cofibrations are the same as in $M$, but the things you want to be generating cofibrations (see Hirschhorn, Chapter 4) only characterize trivial fibrations with fibrant codomain, via lifting. I'd look at examples like $R_K(M)$ to disprove B. Even if all objects of $M$ are fibrant (so that $R_K(M)$ is cofibrantly generated), you've changed $I$ and $W$ without changing $J$, so I think you'll have changed the pseudo-generating sets too. There are also examples where $R_K(M)$ is cofibrantly generated even if $M$ does not have all objects fibrant. I remember Brooke Shipley giving me such an example, where $M$ is sSet. I would also look at examples like this for Question C.