## May 7, 2012

### 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 $\mathcal{M}$:

1. $\mathcal{M}$ is a Cisinski model category, i.e. $\mathcal{M}$ is a Grothendieck topos and its cofibrations are the monomorphisms.
2. $\mathcal{M}$ 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?

Posted at May 7, 2012 9:02 PM UTC

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### 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 exactly if 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.

Posted by: Charles Rezk on May 8, 2012 4:09 AM | Permalink | Reply to this

### Re: The mysterious nature of right properness

The characterization of Charles in particular implies the following:

Given a category $M$ which admits more than one model structure. If one of these model structures is right proper, then all the others are right proper as well. Thus right-properness is really a property of the underlying category with weak equivalences. But as Mike nicely points out in his post, it can of course not be a property of the underlying $\infty$-category.

I found this rather surprising when I heard it for the first time.

Posted by: Thomas Nikolaus on May 8, 2012 9:01 AM | Permalink | Reply to this

### Re: The mysterious nature of right properness

That’s a very nice characterization; thanks Charles! I wonder if that provides any insight into why $sSet_{Quillen}$ is right proper while $sSet_{Joyal}$ isn’t. The model structure $sSet_{Quillen}/S$ is supposed to present the $(\infty,1)$-category $\infty Gpd/S \simeq [S, \infty Gpd]$, which is determined perfectly well once you know a presentation of $S$. But $sSet_{Joyal}/S$ is supposed to present $(\infty,1)Cat/S$, which (at least according to 1-categorical analogy) should be equivalent to lax functors from $S$ into $(\infty,1)Prof$. And a lax functor out of $S$ is not determined by its action on a presentation of $S$, since the image of $g f$ is not in general even equivalent to the composite of the images of $g$ and $f$. Very nice!

It’s also true that the contravariant model structure on $sSet/S$ has the right Quillen equivalence type regardless of whether $S$ is a quasicategory. But the contravariant model structure itself is not right proper. Hmm.

Thus right-properness is really a property of the underlying category with weak equivalences.

Intriguing… now that you say this, it rings a faint bell; maybe I’ve heard this before. Is there a way to characterize right-properness purely in terms of a category with weak equivalences? I guess you could say “any weak equivalence $f\colon x\to y$ induces an equivalence $(C/x)[W^{-1}] \to (C/y)[W^{-1}]$” but that’s not very explicit.

Posted by: Mike Shulman on May 8, 2012 5:34 PM | Permalink | Reply to this

### Re: The mysterious nature of right properness

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.

Let’s see. How about the Strøm model structure?

Posted by: Urs Schreiber on May 8, 2012 7:53 AM | Permalink | Reply to this

### Re: The mysterious nature of right properness

Yes, I thought of that while I was writing the post, but decided not to mention it. The Strøm model structure is hard to get my brain around from a higher-categorical point of view. (-: But $Ho_\infty(Top_{Strom})$ does contain $\infty Gpd$ as a full coreflective subcategory, and any theorem about $(\infty,1)$-categories that aren’t presentable by model categories with all objects bifibrant would have to exclude it somehow. I suppose one way to do that would be that $Ho_\infty(Top_{Strom})$ (probably) isn’t locally presentable as an $(\infty,1)$-category.

Posted by: Mike Shulman on May 8, 2012 5:21 PM | Permalink | Reply to this

### Re: The mysterious nature of right properness

Further on nontrivial examples of model structures with all objects fibrant and cofibrant (just for my own sake, so that I get a feeling for what is possible):

below def. 4.3 in

Apostolos Beligiannis, Homotopy theory of modules and Gorenstein rings, Math. Scand. 89 (2001)

there is a construction of a class of model structures on additive categories with all objects bifibrant. (However, the author seems to relax the conditions on existence of limits and colimits, if I understand correctly.)

This culminates in theorem 4.6 which says that all model structures on additive categories with all objects bifibrant arise this way.

Posted by: Urs Schreiber on May 8, 2012 6:40 PM | Permalink | Reply to this

### Re: The mysterious nature of right properness

Along those lines, there is an analogue of the Strøm model structure on almost any topologically enriched category. This seems a good place to advertise the recent preprint

• Tobias Barthel and Emily Riehl, On the construction of functorial factorizations for model categories, arXiv:1204.5427

which quite neatly fixes an error in Cole’s original construction of such model structures. Intuitively, you want to use the mapping path space for the (acyclic cofibration, fibration) factorization, but the first map in that factorization, though a homotopy equivalence, is not a Hurewicz cofibration (not even in $Top$). Cole’s construction is basically (although he didn’t phrase it in this way) to perform the small object argument but using the mapping path space factorization instead of the usual “one-step” factorization obtained as a pushout of a coproduct of the generators. This doesn’t quite work — the lifting property fails for a subtle reason — but Tobias and Emily showed that it does work if you use the algebraic small object argument. Intuitively, the one-step factorization in the usual small object argument is “free enough” that you can get away with the ordinary SOA, but the mapping path space isn’t.

Tobias and Emily also showed that in $Top$, you can get away with the mapping path space itself if you use Moore paths, which is quite cute.

I find it particularly amusing that this is a place where homotopy type theory is better behaved than classical topology. In classical topology, neither the mapping path space nor the mapping cylinder actually gives you a weak factorization system. (The second map in the mapping cylinder factorization is a homotopy equivalence, but not a Hurewicz fibration.) However, in homotopy type theory, both of them do, and fit together to give a model structure on the category of types (if you forget about the fact that it doesn’t have limits and colimits). For the mapping path space this was proven by Gambino and Garner; for the mapping cylinder (defined as a higher inductive type) it was proven by Peter Lumsdaine.

The type-theoretic “model structure” also has the property that all objects are fibrant and cofibrant. But once you’re willing to relax the requirement of limits and colimits, you can just take the full subcategory of fibrant+cofibrant objects in any model category.

Posted by: Mike Shulman on May 8, 2012 7:15 PM | Permalink | Reply to this

### Re: The mysterious nature of right properness

Along those lines, there is an analogue of the Strøm model structure on almost any topologically enriched category.

Ah, interesting! I need to do something else now, but I have recorded the link here, for later…

But once you’re willing to relax the requirement of limits and colimits, you can just take the full subcategory of fibrant+cofibrant objects in any model category.

Yeah, that article by Beligiannis is a maybe a bit too relaxed in its use of the term “model category”.

Posted by: Urs Schreiber on May 8, 2012 7:40 PM | Permalink | Reply to this

### Re: The mysterious nature of right properness

Of course, in general the full subcategory of bifibrant objects may not admit any limits or colimits. In the type-theoretic “model category”, we have finite products, and pullbacks of fibrations, and it is easy to add type constructors that give us finite coproducts (namely, coproduct types with judgmental $\eta$ conversion).

Posted by: Mike Shulman on May 9, 2012 6:52 AM | Permalink | Reply to this

### Re: The mysterious nature of right properness

Dear Mike, you are right to think that right proper Cisinski model categories correspond to locally presentable $(\infty,1)$-categories which are locally cartesian closed.

A first result in this direction is the following: a Cisinski model structure $\mathcal{M}$ is right proper if and only if, for any fibration between fibrant objects $p:X\to Y$, the pullback functor $p^*:\mathcal{M}/Y\to\mathcal{M}/X$ is a left Quillen functor (this can be deduced straight away from Theorem 4.8 in my paper THT (for Théories homotopiques dans les topos, J. Pure Appl. Algebra 174 (2002), 43-82; a copy is available on my web page)).

As you noticed yourself, any right proper Cisinski model structure $\mathcal{M}$ defines a locally presentable $(\infty,1)$-category $Ho_\infty(\mathcal{M})$ which is locally cartesian. Let us prove the converse.

Let $\mathcal{C}$ be a locally presentable $(\infty,1)$-category. For any big enough regular cardinal $\kappa$, we have that the category $\mathcal{C}_\kappa$ of $\kappa$-compact objects in $\mathcal{C}$ is closed under finite limits and that $\mathcal{C}=Ind_\kappa(\mathcal{C}_\kappa)$ (i.e. any object of $\mathcal{C}$ is a $\kappa$-filtering colimit of $\kappa$-compact objects). Let $C_\kappa$ be a fibrant simplicial category which is a model for $\mathcal{C}_\kappa$. For a simplicial category $A$, I will write $P(A)$ for the category of simplicial presheaves on $A$ endowed with the injective model structure, and $h:A\to P(A)$ for the Yoneda embedding. For a (cofibrant) $\kappa$-small simplicial category $I$ and a simplicial functor $F:I\to C_\kappa$, there is an object $hocolim F$ in $C_\kappa$ which correspond to the colimit of (the nerve of $F$) in $\mathcal{C}_\kappa$ (which always exists, by virtue of Theorem 5.5.1.1 of Lurie’s book), so that we have a canonical map (at least, homotopy theoretically): $(\ast)\qquad hocolim h(F)\to h(hocolim F)$ Let $S$ be the class of maps of shape $(\ast)$. Then the left Bousfield localization $\mathcal{M}$ of $P(C_\kappa)$ by $S$ is a model for $\mathcal{C}$ (whenever $\kappa$ is big enough to imply the identification $\mathcal{C}=Ind_\kappa(\mathcal{C}_\kappa)$). Furthermore, any representable presheaf is $S$-local: indeed, for any object $c$ of $C_\kappa$, and any simplicial functor $F:I\to C_\kappa$ with $I$ $\kappa$-small, the presheaf $Map(-,c)$ sends $hocolim F$ to $holim Map(F,c)$.

Assume from now on that $\mathcal{C}$ is a locally cartesian closed $(\infty,1)$-category. I claim that $\mathcal{M}$ is right proper. Consider a fibration between fibrant objects $p:X\to Y$ of $\mathcal{M}$. Applying Theorem 4.8 in THT in its precise form, it is sufficient to prove that pulling back along $p$ sends any map of the form $(\ast)$ to a weak equivalence.

CLaim 1. For any object $c$ of $C_\kappa$ and any map $s:h(c)\to Y$, the pullback $h(c)\times_Y X$ is an homotopy pullback in $\mathcal{M}$.

To see this, we may factor the map $s$ as a weak equivalence $u:h(c)\to W$ followed by a fibration $q:W\to Y$ in the model category $\mathcal{M}$. The homotopy pullback $h(c)\times^h_Y X$ is then $W\times_Y X$. As the model category $P(C_\kappa)$ is right proper, it is sufficient to prove that $u$ is in fact a weak equivalence of $P(C_\kappa)$ (because any fibration of $\mathcal{M}$ is a fibration of $P(C_\kappa)$): this will imply that the map $h(c)\times_Y X\to W\times_Y X$ is a weak equivalence in $P(C_\kappa)$, whence on $\mathcal{M}$ as well. But $W$ is a fibrant replacement of $h(c)$ in the left Bousfield localization by $S$, and we know that $h(c)$ is $S$-local, from which we deduce that $u$ is a weak equivalence of $P(C_\kappa)$.

Consider now a cofibrant $\kappa$-small simplicial category $I$ and a simplicial functor $F:I\to C_\kappa$ and a map $s:h(hocolim F)\to Y$. For each object $i$ of $I$, we know from Claim 1 that $h(F_i)\times_Y X$ is the homotopy pullback $h(F_i)\times^h_Y X$, which implies that the map $hocolim_{i\in I} (h(F_i)\times_Y X)\to h(hocolim_{i\in I}F_i)\times_Y X$ is a weak equivalence: we are expressing here that pulling back along $p$ preserves homotopy colimits, which is true because $\mathcal{C}$ is assumed to be locally cartesian closed as an $(\infty,1)$-category.

Therefore, to finish the proof, it is sufficient to check

CLaim 2. The canonical map $hocolim_{i\in I} (h(F_i)\times_Y X)\to (hocolim_{i\in I} h(F_i))\times_Y X$ is a weak equivalence.

As homotopy colimits are preserved by left Bousfield localization, it is sufficient to prove Claim 2 in the context of the minimal Cisinski model structure on the topos $P(C_\kappa)/Y$. But then the class of weak equivalences is closed under finite products (see Corollaire (c) of Proposition 3.12 of THT), which implies that the functor $(-)\times_Y X$ is left Quillen, and thus implies CLaim 2.

Finally, here are some remarks about model categories in which any object is both fibrant and cofibrant. It may happend that a model category has this property and is cofibrantly generated. An easy example is the projective model category for complexes of $k$-modules in the case where $k$ is a field. A variation on the same theme is the stable model category of modules over a Frobenius ring (see Theorem 2.2.12 in Hovey’s book) (these two examples fall into the realm of abelian Frobenius categories). There are higher structures there. I guess that the example of Frobenius rings is a little more convincing, but even in the case of complexes over a field $k$, if $k$ is of positive characteristic, there is a non trivial obstruction to strictify an $E_\infty$-algebra into a genuine commutative monoid in the category of $k$-vector spaces, which is the starting point of all the story of the Steenrod algebra. These examples show that the presence of higher phenomena is not directly related to the problem of the existence of a model in which any object is cofibrant and fibrant.

Posted by: Denis-Charles Cisinski on May 8, 2012 10:34 PM | Permalink | Reply to this

### Re: The mysterious nature of right properness

Hi Denis-Charles, nice to see you here again. It would be great if this were true! But I don’t quite follow your proof yet; perhaps there is a typo or I am misunderstanding. It seems to me that your statement of claim 2 is actually the same as the conclusion that we get from knowing that homotopy pullback preserves homotopy colimits. How do we derive any information from this about $h(hocolim_{i\in I} F_i)$?

Posted by: Mike Shulman on May 9, 2012 6:48 AM | Permalink | Reply to this

### Re: The mysterious nature of right properness

(The proof of) Claim 2 says that the non derived pullback functor $(-)\times_Y X$ preserves homotopy colimits up to weak equivalences (although I don’t claim that it preserves weak equivalences of $\mathcal{M}$!).

Therefore, we know that the homotopy colimit of the $h(F_i)\times_Y X$’s is the same as $(hocolim h(F))\times_Y X$. But, by Claim 1, each $h(F_i)\times_Y X$ is the homotopy pullback $h(F_i)\times^h_Y X$ and as homotopy pullback along $p$ preserves homotopy colimits, we have that the homotopy colimit of the $h(F_i)\times^h_Y X$’s is the same as $(hocolim h(F))\times^h_Y X$. But the latter is again the same as $(h(hocolim F))\times^h_Y X$ (because the Yoneda embedding $h:C_\kappa\to\mathcal{M}$ preserves homotopy colimits, by construction of $\mathcal{M}$). Using again Claim 1, we deduce that $(h(hocolim F))\times^h_Y X$ is the same as the non derived pullback $(h(hocolim F))\times_Y X$. In other words, what we have done so far is that we identified both $(hocolim h(F))\times_Y X$ and $(h(hocolim F))\times_Y X$ with $(hocolim h(F))\times^h_Y X$.

Posted by: Denis-Charles Cisinski on May 9, 2012 9:08 AM | Permalink | Reply to this

### Re: The mysterious nature of right properness

Ah, I think I get it! Thanks for explaining further. So the point is that we know that homotopy pullback preserves weak equivalences (in any model category), and so what we need to do is identify the strict pullbacks on both sides with the homotopy pullbacks. Since representables are $S$-local, strict pullbacks of them (along fibrations between fibrant objects) are homotopy pullbacks; this deals immediately with the RHS (the pullback of $h(hocolim_{i\in I} F_i)$) and also with the individual pullbacks of the $h(F_i)$. Thus, since the LHS is a hocolim of these representables, it suffices to know that both strict and homotopy pullbacks preserve homotopy colimits. For homotopy pullbacks this follows from local cartesian closure; for strict pullbacks it follows from your proof of Claim 2. Is that an accurate summary?

Posted by: Mike Shulman on May 9, 2012 11:21 PM | Permalink | Reply to this

### Re: The mysterious nature of right properness

Yes your summary (of the summary I gave of the proof) is accurate.

Posted by: Denis-Charles Cisinski on May 9, 2012 11:32 PM | Permalink | Reply to this

### Re: The mysterious nature of right properness

This is a nice result!

Just for completeness, I have added at locally cartesian closed (∞,1)-category below the theorem a pointer to further equivalent characterizations in HTT.

(Don’t have the energy to do more right now.)

Posted by: Urs Schreiber on May 10, 2012 1:51 AM | Permalink | Reply to this

### Re: The mysterious nature of right properness

Yes… some more trivial examples of this are the canonical model structure on $Cat$, or the trivial model structure on $Set$. (-: But your point is well-taken that even a model category which indisputably represents fully $\infty$-level structure may have this property.