### The Mysterious Nature of Right Properness

#### Posted by Mike Shulman

I’ve been spending too much time recently thinking about (among other things) right properness of model categories. The ultimate goal is to build models of homotopy type theory in $(\infty,1)$-toposes, but at the moment (in this post) I’m just trying to get a handle on about what right properness means, at an intuitive level. So this is going to be kind of rambly and philosophical and lacking in conclusions.

The issue of right properness is an aspect of a more general question: how do properties of a model category — call it $\mathcal{M}$, say — reflect properties of the $(\infty,1)$-category that it presents — call it $Ho_\infty(\mathcal{M})$? I’ve started thinking of this as a sort of *coherence problem*: if we view a model category as a certain sort of “strictification” of an $(\infty,1)$-category, then asking “which $(\infty,1)$-categories can be presented by a model category with such-and-such property?” is the same sort of question as “can every tricategory be presented by a strict 3-category?” In classical coherence theory, we know that sometimes we can make some aspects of a structure strict at the expense of others, e.g. we can make the units in a tricategory strict, or the interchange law, but not both at once. As we’ll see, model categories display similar phenomena.

Recall that a model category $\mathcal{M}$ is called **right proper** if weak equivalences are preserved by pullback along fibrations. Note that this has no immediate interpretation in terms of $Ho_\infty(\mathcal{M})$, since equivalences in an $(\infty,1)$-category are always stable under $(\infty,1)$-pullback; it has more to do with which pullbacks in $\mathcal{M}$ are homotopically meaningful.

Now it’s provable from the axioms of a model category that weak equivalences *between fibrant objects* are preserved by pullback along fibrations, so any model category in which all objects are fibrant is *a fortiori* right proper. Thus, we could start by asking what it means for all objects to be fibrant.

Sometimes, at least, all objects being fibrant means that the objects of $\mathcal{M}$ are fully “algebraic”, in the sense that all “operations” which are supposed to exist in the objects of $Ho_\infty(\mathcal{M})$ already exist in all the objects of $\mathcal{M}$. For instance, all objects of $Cat$ and $Gpd$ are fibrant, because they are algebraic models for higher categories, while not all objects of $sSet_{Quillen}$ or $sSet_{Joyal}$ are fibrant, because they are non-algebraic models for higher categories. Only in the Kan complexes or quasicategories do all the desired “compositions” exist.

Recall also the theorem of Nikolaus that model structures can often be lifted to categories of algebraically fibrant object, thereby making the “operations” in fibrant objects into actual category-theoretic operations. This produces for many $\mathcal{M}$ a Quillen equivalent model category in which all objects are fibrant — so that in particular, all objects being fibrant in $\mathcal{M}$ itself imposes no condition on $Ho_\infty(\mathcal{M})$.

However, this intuition about fibrancy is less obviously correct in other sorts of model categories. One can argue that it works for topological spaces, where the fact that we can directly concatenate paths and homotopies means that “all operations exist”. But for, say, the injective model structure on chain complexes, where the fibrant objects are (roughly) the complexes of injective modules, what are the “operations” that are supposed to exist? Or in the injective model structure on simplicial presheaves? One can give a tautological answer (liftings against all generating acyclic cofibrations), but it doesn’t really help my intuition any. I’d love to hear any better answers.

Of course, the basic theory of model categories is self-dual, but the examples are not. So all objects being *cofibrant* is usually a different sort of condition, in fact almost the dual condition of being fully “non-algebraic”. In an algebraic model, the morphisms have to preserve all the operations strictly, so in order to get weak morphisms we generally need to map out of objects with extra flab — cofibrant objects. If all objects are cofibrant, then that means there are no “operations” that could be preserved more strictly than desired, i.e. we are “fully nonalgebraic”. (In the low dimensions of $Cat$ and $Gpd$, all objects are cofibrant for the different reason that there is no room for a difference between strictness and non-strictness of morphisms.)

From this point of view, it seems that a nontrivial higher structure probably cannot be presented by any model category in which all objects are *both* fibrant and cofibrant. I would be very surprised if $\infty Gpd$ could be presented by such a model category. I would be fairly surprised if even $2 Gpd$ could be! But I don’t know of *any* $(\infty,1)$-category that *provably* cannot be presented by such a model category. Nor do I have much of an idea how one could go about proving such a thing. What could knowing that all objects of $\mathcal{M}$ are fibrant and cofibrant tell us about $Ho_\infty(\mathcal{M})$? Any ideas?

Moving from a smaller mystery to a greater one, I suggested above that right properness could be thought of as a sort of weaker version of all objects being fibrant. If a model category fails to be right proper, then we can often give an explicit counterexample, which seems to say that the problem is that the objects are “not fibrant enough”. For instance, $sSet_{Joyal}$ is not right proper: the pullback of the inner horn $\Lambda^2_1 \hookrightarrow \Delta^2$ along the Joyal fibration $d_1\colon \Delta^1 \to \Delta^2$ is $\partial\Delta^{1} \hookrightarrow \Delta^1$, which is no longer a weak equivalence. “Obviously” the problem is that $\Lambda^2_1$ is “very not-fibrant”.

However, $sSet_{Quillen}$ *is* right proper! Is right properness telling us that arbitrary simplicial sets are somehow closer to being Kan complexes than they are to being quasicategories? That would seem an odd thing to say. And if we write down the same example, then $\Lambda^2_1$ is no more a Kan complex than it is a quasicategory, but now $d_1\colon \Delta^1 \to \Delta^2$ is not a Kan fibration. So does right properness mean that there are not too many fibrations relative to the fibrant objects? Maybe right properness means different things in different model categories.

Let me end with the example that I care most about, and which I find the most mysterious. Consider the following two properties of a model category $\mathcal{M}$:

- $\mathcal{M}$ is a Cisinski model category, i.e. $\mathcal{M}$ is a Grothendieck topos and its cofibrations are the monomorphisms.
- $\mathcal{M}$ is right proper.

By a theorem of Dugger, we know that every locally presentable $(\infty,1)$-category can be presented as a localization of a model category of simplicial presheaves. If we perform this localization on the injective model structure, we obtain a Cisinski model for any locally presentable $(\infty,1)$-category. Thus, the first condition imposes basically no condition on $Ho_\infty(\mathcal{M})$.

We have seen that the third condition also imposes basically no condition on $Ho_\infty(\mathcal{M})$. In fact we can almost always find a model for $Ho_\infty(\mathcal{M})$ in which all objects are fibrant, say by starting with a Cisinski model category and passing to algebraically fibrant objects.

However, if we ask for *both* properties at once, something magical happens. Since the cofibrations are the monomorphisms, they are stable under pullback. Since $\mathcal{M}$ is right proper, therefore, acyclic cofibrations are stable under pullback along fibrations. And since $\mathcal{M}$ is locally cartesian closed (being a topos), any morphism $g\colon A\to B$ induces an adjunction $g^\ast : \mathcal{M}/B \rightleftarrows \mathcal{M}/A : \Pi_g$, and we have just observed that when $g$ is a fibration, this adjunction is Quillen. It follows that the derived version of $g^\ast$, from $Ho_\infty(\mathcal{M})/B$ to $Ho_\infty(\mathcal{M})/A$, has a right adjoint, and so $Ho_\infty(\mathcal{M})$ is locally cartesian closed as an $(\infty,1)$-category.

Needless to say, not every locally presentable $(\infty,1)$-category is locally cartesian closed. But is this the only condition? Can any locally cartesian closed, locally presentable $(\infty,1)$-category be presented by a right proper Cisinski model category?

## Re: The mysterious nature of right properness

I like this characterization of right properness:

Given a model category $M$, every slice category $M/X$ gets a model category structure in a standard way, and to every map $f:X\to Y$ there is a standard Quillen pair $M/X \rightleftarrows M/Y$. In general, if the standard Quillen pair $M/X\rightleftarrows M/Y$ associated to $f$ is a Quillen equivence, then $f$ has to be a weak equivalence.

$M$ is right proper

exactlyif the converse holds: i.e., if $f$ a weak equivalence implies that $M/X\rightleftarrows M/Y$ is a Quillen equivalence.If $X$ and $Y$ happen to be fibrant, then the converse statement will hold. It is only for general objects that a problem arises.

In other words, $M$ is right proper iff all slice categories have the “correct” Quillen equivalence type; i.e., if $X\mapsto h(M/X)$ coincides with its derived functor. So I guess you could say an object $X$ is “quasi-fibrant” if $M/X$ has the correct Quillen equivalence type, so that right proper model categories are the ones with all objects quasi-fibrant. I don’t know if this is the sort of thing you are after.