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Quasi-categories (or $\infty$-categories, as they are often called) are a very convenient setting for doing abstract homotopy theory. One of their amazing features is the following: Given a diagram of quasi-categories, we can form its homotopy limit, yielding a quasi-category again. For example, the inverse (homotopy) limit of the diagram

$\cdots \xrightarrow{\Omega} \mathcal{S}_* \xrightarrow{\Omega} \mathcal{S}_* \xrightarrow{\Omega} \mathcal{S}_*$

(for $\mathcal{S}_*$ the quasi-category of pointed spaces and $\Omega$ denotes the loop space) gives the quasi-category of spectra. These homotopy limits can be (abstractly) defined to be the homotopy limits in the Joyal model structure on simplicial sets, where the quasi-categories are just the fibrant objects. There is also a more explicit description given in Lurie's Higher Topos Theory (we come back to an example later in this question).

Many important examples of quasi-categories are constructed from a simplicial model category $\mathcal{M}$ in the following way: The sub simplicial category $\mathcal{M}^\circ$ of bifibrant objects forms a Bergner fibrant simplicial category and taking the coherent nerve produces a quasi-category $N(\mathcal{M}^\circ)$. Thus, the following question seems to be natural: Can we reconstruct the homotopy limit of the coherent nerves of a diagram of model categories as the coherent nerve of a "homotopy limit" of model categories?

A candidate is given in Julie Bergner's paper Homotopy limits of model categories and more general homotopy theories, Definition 3.1. We won't recall here the general definition, but only indicate it in the case that we index over a diagram with one object and a group $G$ as automorphism (i.e., we have a group action on our model category): Then an object in $holim_G \mathcal{M}$ is an object $X \in \mathcal{M}$ together with morphisms $f_g: X \to g\cdot X$ (for $g\in G$) such that $f_e = id_X$ and $f_{hg} = (h\cdot f_g)\circ f_h$ (i.e, objects with a twisted $G$-action). At least in this case, the homotopy limit has a model structure (Bergner mentions the injective one, but at least sometimes, it has also the projective one, which is Quillen equivalent); actually it is a simplicial one if $\mathcal{M}$ was a simplicial model category. Indeed, it is the simplicial subcategory of $G$-equivariant morphisms in $Fun(EG, \mathcal{M})$ where $EG$ denotes the contractible groupoid associated to $G$. Thus, more precisely, our question is:

Is $N((holim_G \mathcal{M})^\circ)$ categorically equivalent to $holim_G N(\mathcal{M}^\circ)$ as quasi-categories?

There are several pieces of evidence for this:

  1. If I am not mistaken, the description of homotopy limits in Higher Topos Theory implies that the homotopy fixed points of a quasi-category $\mathcal{C}$ are given as $Map(N(EG), \mathcal{C})^G$ (where Map denotes the internal Hom of simplicial sets and $()^G$ denotes strict fixed points). Thus the question is equivalent to whether $Map(N(EG), N(\mathcal{M}^\circ))^G$ is categorically equivalent to $N((Fun(EG, \mathcal{M})^G)^\circ)$. By strictification of homotopy coherent diagrams, a similar statement holds if we don't take $G$-fixed points, but I don't see how to prove the statement involving the $G$-fixed points.

  2. Even more convincingly, Julie Bergner shows in her paper that homotopy limits of model categories are compatible with homotopy limits in the complete Segal space model structure on simplicial spaces. More precisely, she shows that the classification diagram functor commutes with homotopy limits up to weak equivalence (Theorem 4.1). Now, one could get the impression that we are finished since the complete Segal space model structure and the Joyal model structure are Quillen equivalent. But this is not sufficient: one has to prove that the classification diagram functor is send under this Quillen equivalence to something weakly equivalent to the coherent nerve. Although one gets some compatibility results from the papers Quasi-categories vs. Simplicial Categories (by Andre Joyal) and Complete Segal spaces arising from simplicial categories (by Julie Bergner), I didn't quite find the right statement to make the comparision work.

As a last word of motiviation, I want to add that I stumbled upon these questions when I thought about Galois descent, where one often considers objects with twisted group actions.

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I am guessing that $\Omega$ means loop space? –  Spice the Bird Apr 24 '12 at 14:06
May be I'm wrong, but it seems to me that it should be: $Map(N(EG),N\mathbf({\mathcal{M}}^{\circ}))^{G}$ equivalent to $N((RHom(EG,\mathcal{M})^{G})^{\circ})$, where RHom is the right derived internal Hom functor defined by B.Toën for $\mathrm{Ho}(\mathbf{Cat}_{\Delta})$. –  Fedotov Apr 24 '12 at 14:55
In general, if we have a (say, presentable, or simplicially enriched) model category, then a (co)limit (in the $\infty$-categorical sense) of a diagram in the underlying infinity-category corresponds to a homotopy (co)limit in the model category. Since the category of simplicially enriched categories with the Bergner model structure is Quillen equivalent to the category of simplicial sets with the Joyal model structure, the underlying $\infty$-categories are equivalent. So an $\infty$-categorical (co)limit of $\ifnty$-categories corresponds to a homotopy colimit of simplicial sets which –  Dylan Wilson Apr 24 '12 at 17:19
(contd) corresponds to a homotopy (co)limit of simplicially enriched categories. –  Dylan Wilson Apr 24 '12 at 17:20
Somewhere in there lies an answer to your question, since at the end of the day a homotopy limit of the simplicially enriched model categories will correspond to a homotopy limit of some ordinary simplicially enriched categories... But I may be spouting nonsense. –  Dylan Wilson Apr 24 '12 at 17:21
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1 Answer

up vote 5 down vote accepted

I will address your second question: "one has to prove that the classification diagram functor is sent under this Quillen equivalence to something weakly equivalent to the coherent nerve".

The answer is yes, this is true. First note that the simplicial category $M^\circ$ is equivalent as a simplicial category to $L^HM$, the hammock localization of M (at the weak equivalences). See here for references. So the answer to your second question (or is it a remark?) follows as a special case of this previous MO answer (which cites work of Barwick-Kan and Toen). So the classification diagram functor and the coherent nerve of $M^\circ$ are sent to equivalent things under the Quillen equivalence between quasicategories and complete Segal spaces (I prefer the term Rezk categories or Rezk $\infty$-categories).

I believe that combined with your other comments (notably using Julie Bergener's results) this is enough to answer your main question in the affirmative.

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This seems to work (with a few details added). Many thanks! –  Lennart Meier May 8 '12 at 22:19
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