homotopy limits of dg categories

The question is related to the following MO question

(Co-)Limits and fibrations of DG-Categories?

My question is,

  1. how to define the homotopy limit (and colimit) of a system of dg-categories (let’s fix a universe and a base ring $k$, and work only with small things…), and

  2. is there an explicit description of the homotopy category of the homotopy limit of dg categories
    $$
    Ho(holim_{i\in I}\mathscr C_i)=?
    $$
    Recall that the homotopy category $Ho(\mathscr C)$ of a dg category $\mathscr C$ is the category with the same objects as $\mathscr C$ and the hom group is the cohomology at degree 0 of the hom complex in $\mathscr C:$
    $$
    Hom_{Ho(\mathscr C)}(X,Y)=H^0(Hom_{\mathscr C}(X,Y)).
    $$
    One can ask similar questions to “categories” enriched in simplicial sets, which is a slightly more general setting.

I understand (sort of) that there is a model category structure (due to Tabuada) on the category $dg-Cat$ of dg categories such that weak equivalences are what one expects (to be a bit precise, a functor $F:\mathscr C\to\mathscr D$ is a w.e. if
$$
Hom_{\mathscr C}(X,Y)\to Hom_{\mathscr D}(FX,FY)
$$
is a quasi-isomorphism of complexes, and $Ho(F):Ho(\mathscr C)\to Ho(\mathscr D)$ is essentially surjective). But I don’t know how to use this model structure to define homotopy limits.

Maybe one uses cofibrant replacement and the naive $\otimes$-structure on $dg-Cat$ to define a $\otimes^{\mathbb L}$-structure (following Toen) and shows that it is closed, so that one has internal hom $R\mathscr Hom$ on $dg-Cat,$ with which one defines homotopy limits (and colimits) of dg-categories by universal properties. I’m not sure. Both references and direct explanations are appreciated.

Answer

Let me sketch the definition of homotopy limit in full generality. Suppose $\mathscr{M}$ is a category with weak equivalences. Denote $\operatorname{Ho}(\mathscr{M})$ the category obtained by inverting weak equivalences. For any small category $I$, denote $\mathscr{M}^I$ the category of functors $I\rightarrow \mathscr{M}$. Defite weak equivalences in $\mathscr{M}^I$ to be the natural transformations between functors whose values are weak equivalences in $\mathscr{M}$. The ‘constant diagram’ functor $\mathscr{M}\rightarrow \mathscr{M}^I$ preserves weak equivalences, therefore it defines a functor,
$$\text{constant diagram}\colon \operatorname{Ho}(\mathscr{M})\longrightarrow \operatorname{Ho}(\mathscr{M}^I).$$
The homotopy limit functor,
$$\operatorname{holim}_{i\in I}\colon \operatorname{Ho}(\mathscr{M}^I)\longrightarrow \operatorname{Ho}(\mathscr{M}),$$
if it exists, is the right adjoint of the previous functor.

Notice that homotopy limits depend on the weak equivalences we consider. You have mentioned one of the weak equivalences you can take on DG-categories. There are other very interesting weak equivalences that you could also consider, and that would yield different homotopy limits.

The model category techniques show the existence of homotopy limits under certain hypotheses and tell us how to construct them from resolutions. A good reference is:

MR1944041 (2003j:18018) Hirschhorn, Philip S. Model categories and their localizations. Mathematical Surveys and Monographs, 99. American Mathematical Society, Providence, RI, 2003. xvi+457 pp. ISBN: 0-8218-3279-4 (Reviewer: David A. Blanc), 18G55 (55P60 55U35)

Suppose now that $\mathscr{M}$ is the model category of DG-categories you consider, and let $\mathscr{N}$ be the model category of $k$-linear categories where weak equivalences are $k$-linear equivaleces of categories. We can regard any $k$-linear category as a DG-category concentrated in degree $0$. The inclusion $\mathscr{N}\subset\mathscr{M}$ is a left Quillen functor with right adjoint

$$H^0\colon\mathscr{M}\longrightarrow\mathscr{N}.$$

This and the uniqueness of adjoints can be used to show that

$$H^0(\operatorname{holim}_{{i\in I}}\mathscr{C}_i) = \operatorname{holim}_I H^0(\mathscr{C}_i)$$

Here the second homotopy limit is in $\mathscr{N}$ which is hopefully easier to compute since the model category structure on ${\mathscr{N}}$ is simpler. I guess that further simplifications depend on particular cases you may want to consider.

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