The n-Category Café

In the following we will relate right properness of model structures obtained via left Bousfield localization with semi-left exactness of the Bousfield localizations. Therefore, we give a definition of semi-left exact left Bousfield localizations and, while at it, also define combinatorial model categories with universal homotopy colimits to build a purely model categorical framework in which to phrase our observations in. A more detailed account can be found in Chapter 7 of my PhD thesis, prepared at the University of Leeds under the supervision of Nicola Gambino and submitted last month.

Recall that by Dugger’s and Lurie’s work there is a correspondence between combinatorial model categories and presentable $(\infty,1)$-categories, such that left Bousfield localizations of the former translate to reflective localizations of the latter.

The setting of model toposes developed by Rezk, and Toen and Vezzosi, and the setting of Grothendieck $\infty$-toposes developed by Lurie translate as a particular class of combinatorial/presentable objects back and forth in an analogous fashion. Indeed, in a nutshell, by the work of Rezk (Toposes and Homotopy Toposes) model toposes can be described exactly as those combinatorial model categories which are Quillen equivalent to left exact left Bousfield localizations of simplicial presheaf categories $\mathrm{sPsh}(\mathbb{C})_{\mathrm{proj}}$ for small simplicial categories $\mathbb{C}$. These are presentations of left exact localizations of presheaf $\infty$-toposes in the sense of Lurie. And hence emerges the correspondence of model toposes and Grothendieck $\infty$-toposes.

Now, the class of presentable locally cartesian closed $(\infty,1)$-categories and their relation to semi-left exact localizations has been analyzed by Gepner and Kock (Univalence in locally cartesian closed $\infty$-categories, arxiv). Here, a localization $L\colon\mathcal{C}\rightarrow\mathcal{L}\mathcal{C}$ is said to be semi-left exact if $L$ preserves pullbacks of spans $(f\colon a\rightarrow b, g\colon c\rightarrow b)$ where $a$ and $b$ are local objects.

In the spirit of Rezk’s approach, in order to lift this analysis to a purely model categorical setting, say that a combinatorial model category has universal homotopy colimits if homotopy colimits commute with homotopy pullbacks in the sense of Toen and Vezzosi’s Giraud Axiom given in Definition 4.9.1.2 of HAG I. Then one can show that a combinatorial model category $\mathcal{M}$ has universal homotopy colimits iff any of the following conditions hold.

• $\mathcal{M}$ satisfies Toen and Vezzosi’s Definition 4.9.1.2 for all arrows with or without fibrant domain and codomain.
• $\mathcal{M}$ satisfies Rezk’s descent property (P1) defined in Section 6.5 of Toposes and Homotopy
  Toposes.
• $Ho_{\infty}(\mathcal{M})$ is locally cartesian closed.

Then, we say that a left Bousfield localization $L\colon\mathcal{M}\rightarrow\mathcal{L}\mathcal{M}$ is semi-left exact if $L$ preserves the homotopy pullback of spans $(f\colon A\rightarrow B, g\colon C\rightarrow B)$ such that the objects $A,B$ are fibrant in $\mathcal{L}\mathcal{M}$.

We hence obtain a correspondence between semi-left exact left Bousfield localizations between combinatorial model categories and semi-left exact localizations of presentable $(\infty,1)$-categories in the sense of Gepner and Kock. It then follows that, if $\mathcal{M}$ is combinatorial with universal homotopy colimits, then semi-left exactness of $L\colon\mathcal{M}\rightarrow\mathcal{L}\mathcal{M}$ implies that $\mathcal{L}\mathcal{M}$ has universal homotopy colimits, too. It is reasonable to expect the converse to hold under the condition that the reflection $R$ preserves dependent products (as it is the case in the 1-categorical setting).

This yields a presentation theorem for combinatorial model categories with universal homotopy colimits via semi-left exact localizations of simplicial presheaf categories by the first part of Rezk’s proof of Theorem 6.9 in Toposes and Homotopy Toposes (that is the corresponding statement for model toposes and the presentation of such via left exact localizations).

Now, to eventually come back to right properness, we observe the following two facts. Let $\mathcal{M}$ be a model category and $\mathcal{M}\rightarrow\mathcal{L}\mathcal{M}$ a left Bousfield localization (note that we do not make any assumptions on $\mathcal{M}$).

1. Suppose $\mathcal{L}\mathcal{M}$ is right proper. Then the Bousfield localization $\mathcal{M}\rightarrow\mathcal{L}\mathcal{M}$ is semi-left exact.
2. Suppose $\mathcal{M}$ is right proper. Then the Bousfield localization $\mathcal{M}\rightarrow\mathcal{L}\mathcal{M}$ is semi-left exact if and only if the model category $\mathcal{L}\mathcal{M}$ is right proper.

Proof: For arrows $f\colon A\rightarrow B$ and $g\colon C\rightarrow B$ in $\mathcal{M}$, successively replacing the objects and arrows fibrantly first in $\mathcal{M}$ and then in $\mathcal{L}\mathcal{M}$ gives the following sequence of pullback squares.

For Fact 1, assume $\mathcal{L}\mathcal{M}$ is right proper and let $f\colon A\rightarrow B$ be a map between local objects and $g\colon C\rightarrow B$. Here, in this generality by “local” we simply mean that $A$ and $B$ are fibrant objects in $\mathcal{L}\mathcal{M}$. Then, in the diagram, the fibrant replacements $\mathbb{R}A$ and $\mathbb{R}B$ are local, too, that means in fact already fibrant in the localization $\mathcal{L}\mathcal{M}$. So the fibration $\mathbb{R}f$ is also a fibration in $\mathcal{L}\mathcal{M}$. But this implies that the map $Q\rightarrow S$ is a weak equivalence in $\mathcal{L}\mathcal{M}$, because $\mathcal{L}\mathcal{M}$ is right proper.

For Fact 2, suppose the localization is semi-left exact, let $f\colon A\rightarrow B$ be a fibration and $g\colon C\rightarrow B$ be a weak equivalence in $\mathcal{L}\mathcal{M}$. Without loss of generality we can assume that $B$ is fibrant in $\mathcal{L}\mathcal{M}$. Then the map $P\rightarrow Q$ is a weak equivalence in $\mathcal{M}$ by right properness of $\mathcal{M}$ and fibrancy of $f$. Also, because $B$ was assumed to be local, so are $\mathbb{R}B$ and hence $\mathbb{R}A$, thus the map $Q\rightarrow S$ is a weak equivalence by semi-left exactness of the localization. So all diagonal arrows in the diagram are weak equivalences in $\mathcal{L}\mathcal{M}$. Since $g$ was assumed to be a weak equivalence, by 2-for-3, the map $\mathbb{R}_L g$ is an acyclic fibration. Therefore, so is $S\rightarrow \mathbb{R}_L A$. But then, again by 2-for-3, the map $f^{\ast}g\colon P\rightarrow A$ is a weak equivalence in $\mathcal{L}\mathcal{M}$.

The other direction follows immediately from Fact 1.

A few comments on Facts 1 and 2.

• Fact 2 was also observed by Balchin and Garner in Bousfield localisation and colocalisation of one-dimensional model structures for 1-dimensional model categories. Their lemma is a special case as 1-dimensional model categories are always right proper.
• Fact 2 is intuitive in the sense that right properness assures that ordinary pullbacks along fibrations between fibrant objects are homotopy pullbacks, and both model categories $\mathcal{M}$ and $\mathcal{L}\mathcal{M}$ have the same underlying ordinary categorical structure. Fact 1 on the contrary states no compatibility conditions between 1-categorical and higher categorical structure in $\mathcal{M}$ or $\mathcal{L}\mathcal{M}$ and neither does it state any conditions which relate right properness and universal homotopy colimits.
• We have argued above that semi-left exactness and universality of homotopy colimits in the localized model structure are equivalent under some conditions, and it is interesting to note that a similar but stronger relationship between semi-left exactness and right properness is given by Facts (1) and (2).
• In the last post it was noted that every locally cartesian closed $(\infty,1)$-category is presented by a right proper Cisinski model category, obtained as left Bousfield localization of a simplicial presheaf category equipped with the injective model structure. But we note that the connection between right properness and local cartesian closedness in this case is in fact a connection between right properness and semi-left exactness of the localization. The two latter properties just are equivalent in this case.

So we see that right properness of such “standard” presentations $\mathcal{L}_T(\mathrm{sPsh}(\mathbb{C}))$ is not a peculiarity of Cisinskiness, since a simplicial presheaf category equipped with any model structure $\mathcal{M}$ with pointwise weak equivalences is right proper (thanks to Karol Szumilo for making me aware of this), and hence so is any model category obtained from $\mathcal{M}$ by semi-left exact localization.

Recalling that Dugger observed that the projective model structure enjoys a “cofibrancy” status that the injective model structure generally does not, we can vary the observation from the last post (which also was rigorously stated in Theorem 7.1 of Gepner and Kock’s paper) as follows.

Let $\mathcal{M}$ be a combinatorial model category. Then there is a simplicial category $\mathbb{C}$, a set $T\subset \mathrm{sPsh}(\mathbb{C})$ of maps and a Quillen equivalence $$\mathcal{L}_T(\mathrm{sPsh}(\mathbb{C}))_{\mathrm{proj}}\simeq\mathcal{M}$$ such that $\mathcal{L}_T\mathrm{sPsh}(\mathbb{C})$ is right proper if and only if $\mathcal{M}$ has universal homotopy colimits.

As a corollary we see that right properness is homotopy invariant among such “standard” presentations of combinatorial model categories. More precisely, for $\mathbb{C},\mathbb{D}$ small simplicial categories, let $\mathrm{sPsh}(\mathbb{C})$ and $\mathrm{sPsh}(\mathbb{D})$ be equipped with any model structure with pointwise weak equivalences. Let $F\colon\mathrm{sPsh}(\mathbb{C})\simeq\mathrm{sPsh}(\mathbb{D})$ be a Quillen equivalence and $S,T$ be sets of maps such that $F$ descends to a Quillen equivalence $F\colon\mathcal{L}_S\mathrm{sPsh}(\mathbb{C})\simeq\mathcal{L}_T\mathrm{sPsh}(\mathbb{D})$. Then $\mathcal{L}_S\mathrm{sPsh}(\mathbb{C})$ is right proper iff $\mathcal{L}_T\mathrm{sPsh}(\mathbb{D})$ is right proper.