Does $\infty$-categorical localization commute with taking directed fibered products?

Question

Suppose we are given categories $\mathsf{C},\mathsf{D},\mathsf{E},$ equipped with collections of weak equivalences $\mathcal{W}_{\mathsf{C}},\mathcal{W}_{\mathsf{D}},$ and $\mathcal{W}_{\mathsf{E}},$ respectively. Suppose that we are also given functors $F : \mathsf{C}\to\mathsf{E}$ and $G : \mathsf{D}\to\mathsf{E}$ which send weak equivalences to weak equivalences. Denote by $\mathsf{C}\overset{\rightarrow}{\times}_{\mathsf{E}}\mathsf{D}$ the "directed fiber product" (i.e., the comma category of triples $(c\in\mathsf{C},d\in\mathsf{D},\alpha : F(c)\to G(d)\in\mathsf{E})$). We may equip $\mathsf{C}\overset{\rightarrow}{\times}_{\mathsf{E}}\mathsf{D}$ with a collection of weak equivalences $\mathcal{W}$ given by morphisms $$(f,g) : (c,d,\alpha)\to(c',d',\alpha')$$ such that $f\in\mathcal{W}_{\mathsf{C}}$ and $g\in\mathcal{W}_{\mathsf{D}}.$ Let $L(\mathsf{C},\mathcal{W}_{\mathsf{C}})$ denote the $\infty$-categorical localization/underlying $\infty$-category of $\mathsf{C}$ with respect to $\mathcal{W}_{\mathsf{C}}$ (and similarly for the other categories).

My Question: Is there an equivalence of $\infty$-categories $$ L(\mathsf{C},\mathcal{W}_{\mathsf{C}})\overset{\rightarrow}{\times}_{L(\mathsf{E},\mathcal{W}_{\mathsf{E}})}L(\mathsf{D},\mathcal{W}_{\mathsf{D}})\simeq L(\mathsf{C}\overset{\rightarrow}{\times}_{\mathsf{E}}\mathsf{D},\mathcal{W}), $$ and if not, are there any reasonable conditions under which this would hold?

I'm happy to assume that all of the categories in question are proper, simplicial, cellular model categories and that the weak equivalences are the weak equivalences in these model structures, and even that $\mathsf{C} = \mathsf{E},$ $\mathcal{W}_{\mathsf{C}} = \mathcal{W}_{\mathsf{E}},$ and that $F = \operatorname{id}.$

I'm aware that this holds for $1$-categorical localizations of products of model categories, as in this question, and I presume the statement for products also holds when we consider $\infty$-categorical localizations of the model categories as well via a similar argument (although I haven't checked this carefully -- please correct me if I'm wrong on this point).

My motivation for this question is that I have encountered a model category that I'm interested in, described as a "directed fiber product" with a model structure as in the question. I want work with the underlying $\infty$-category of this model category, but it would be much easier to do so if I know that I can work with $\infty$-categories from the start to build the category I'm interested in, and avoid the model-category details as much as possible.

Answer 1

Here is a counter example in the general case:

Take $E = [1] = 0 \to 1 $. $D = \{0\} \coprod \{1\} $ and $C = \{1\}$. where all maps are weak equivalence.

The lax-pullback is $\{id:1 \to 1\}$, and the localization of $E$ and $C$ are both the terminal category, while the localization of $D$ is $D$ itself. so the pullback of the localization has two non isomorphic objects, while the pullback only has one objects.

A special case of that question that is well studied is when all maps are weak equivalences and one take a pseudo-pullback. This corresponds to whether a homotopy pullback in the Joyal Model structure is preserved by the localization corresponding to Kan-Quillen model structure.

This is not the case in general, in fact, it is not very often the case. Typical condition under which this is true are given by Quillen's Theorem B, or its generalization to infinity category ( and directly stated in terms of pullback square) which you can find as theorem 4.6.11 of Cisinski "Higher categories and homotopical algebra" (see condition (iii)). One can deduce result about lax pullback from this as well.

I would expect a similar result for general localization might be possible, but I don't think I have even seen it.

Another situation where a positive answer in this direction is possible, is when all categories $E,C$ and $C$ are Brown categories of cofibrant objects, the functor are morphisms of categories of cofibrant objects, the pullback is a pseudo-pullback and one of the functor is a "fibration" in the sense of Karol Szumilo work. Because Szumilo shows that Brown categories of cofibrants objects form a category of fibrant objects, pullback along fibration are homotopy pullback and one can deduce the desired result from this. I would expect one can say something about lax pullback from this as well, but I don't think this has been investigated. Also in your setting one would need to assume that $G$ (and $F$) are left or right Quillen functor to hope to apply this kind of idea...

An alternative version of Szumilo's construction can also be find in my own work, where the notion of "fibration" is a little more general. Though with this version it has not been clearly proved that there is an equivalence with an appropriate class of $(\infty,1)$-categories, so it wouldn't be possible to deduce form the homotopy pullback something about a pullback of $\infty$-categories. Of course, this is also true, it would just require some extra work.