Say $$\mathcal{C'}\to \mathcal{C}\leftarrow \mathcal{D}$$ is a diagram of model categories and (e.g. Left) Quillen functors. I want to write down a (hopefully simple) model category $\mathcal{D}'$, or at least a category with weak equivalences, such that its $\infty$-categorical localization is the homotopy limit of the localizations of this diagram in the $(\infty,1)$ category of $(\infty, 1)$ categories. Is there a nice way to do this? I'm willing to impose any reasonable niceness conditions on the categories in the diagram.


Philippe Gaucher is right. This problem was solved by Julie Bergner, here. I recently asked a question that summarized some of her work on this problem. The point is that the homotopy limit of your diagram is a category $M$ whose objects are 5-tuples $(x_1,x_2,x_3,u,v)$ with $x_1 \in C'$, $x_2 \in D$, $x_3\in C$, and $F(x_1) \stackrel{u}{\to} x_3 \stackrel{v}{\gets} G(x_2)$ in $C$, where $F$ and $G$ are the two functors in your diagram. The morphisms in this category of 5-tuples are obvious. This category $M$ can be given a model structure where the weak equivalences and cofibrations are levelwise (on each $x_i$), and that model structure can be localized if desired to force $u$ and $v$ to be weak equivalences in the local objects of $M$. Bergner then proves $M$ has the correct homotopy type, meaning that, upon passage to complete Segal spaces (i.e. $(\infty,1)$-categories), it becomes the actual homotopy pullback of the diagram. She has to assume the model categories she starts with are combinatorial, but this seems a standard assumption now from the $\infty$-categorical perspective (i.e. assuming presentability). Bergner uses a right Bousfield localization, so you need to assume right properness, or pass to right semi-model categories like Barwick does in this paper. The difference between a semi-model structure and a full model structure is invisible to the underlying $(\infty,1)$-category.

EDIT (in answer to comments): Bergner uses the notation $L_DX$ for the category I called $M$ above. It's the lax homotopy limit. The homotopy limit is the full subcategory where the maps $u$ and $v$ have been forced to be weak equivalences. She does not claim it has a model structure in general, but it does in some special cases, e.g. if $L_DX$ is right proper and combinatorial. This occurs if each of your categories $C,C',D$ is combinatorial and has all objects fibrant, for example. This assumption can be avoided (and Bergner points this out, right after Theorem 3.2 in the linked paper) by using Barwick's method of right Bousfield localization without right properness. The result is a right semi-model structure on $Lim_DX$, and such categories have associated $(\infty,1)$-categories just like model categories do. And Bergner proves that the associated $(\infty,1)$-category is the homotopy limit in the category of $(\infty,1)$-categories, as you'd expect (working in the model of Complete Segal Spaces).

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  • $\begingroup$ Where exactly in her paper does Julie Bergner prove that “This category M can be given a model structure where the weak equivalences and cofibrations are levelwise (on each x_i)”? In Theorem 3.2 she assumes that “L_D X has the structure of a right proper model category” without giving any indications as to its existence. $\endgroup$ – Dmitri Pavlov Jun 20 '18 at 17:40
  • $\begingroup$ This is a little surprising: it sounds like you're describing the Grothendieck construction, which is some kind of (infinity, 2)-categorical pullback, and not the (infinity, 1) categorical pullback. Are you sure your u, v are not required to be weak equivalences of some kind? $\endgroup$ – Dmitry Vaintrob Jun 20 '18 at 19:05
  • $\begingroup$ @DmitriPavlov: She gives the indications about existence directly after finishing the proof of Theorem 3.2. Her first sentence there is "Of course, the di±culty in using this theorem lies in the difficulty in establishing that the model category $L_DX$ is right proper." $\endgroup$ – David White Jun 21 '18 at 0:57
  • $\begingroup$ @DmitryVaintrob: u and v are forced to be weak equivalences once we pass to cofibrant objects. $\endgroup$ – Dmitri Pavlov Jun 21 '18 at 2:26
  • $\begingroup$ @DavidWhite: She says there: “In practice, when the conditions of this theorem cannot be verified, we can still use the original levelwise model structure on L_D X and simply restrict to the appropriate subcategory when we want to require u and v to be weak equivalences.” I fail to see how such a claim could possibly solve the problem of constructing the pullback as a model category. $\endgroup$ – Dmitri Pavlov Jun 21 '18 at 2:31

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