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(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.

  • 1
    $\begingroup$ Maybe section 10 of nd.edu/~wgd/Dvi/Homotopy.Theories.dvi is a good place to start. It explains how to use model structures to produce some easy homotopy limits. $\endgroup$ Apr 27, 2011 at 20:49
  • $\begingroup$ oops...can't open dvi file in my computer...Do you know if there's a pdf or can you tell me the title/reference of the paper? Thanks, Timo. $\endgroup$
    – shenghao
    Apr 27, 2011 at 21:19
  • $\begingroup$ It's Dwyer/Spalinski, "Homotopy theories and model categories". $\endgroup$ Apr 28, 2011 at 8:01
  • 1
    $\begingroup$ And here's a pdf: hopf.math.purdue.edu//Dwyer-Spalinski/theories.pdf $\endgroup$ Apr 28, 2011 at 8:03
  • $\begingroup$ It helps to clarify many things for myself. Thanks again. But still, in the potential application I have in mind, the index $I$ is infinite, at least countably infinite, and in 10.13 only the cases when $I$ is "very small" are considered and the existence of homotopy limits/colimits are proved. Anyway, for general index $I,$ the reference you provided gives the definition, and it's much more accessible for me than Hirschowitz-Simpson's "Descente pour les n-champs". This answers my 1st question. $\endgroup$
    – shenghao
    Apr 28, 2011 at 12:56

2 Answers 2


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


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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    $\begingroup$ Unfortunately, the inclusion of linear categories into dg categories is not a left Quillen functor at all! $\endgroup$ Apr 28, 2011 at 1:32
  • $\begingroup$ @Denis-Charles: oh, you're right! I forgot about cofibrations... Could it still be possible that $H^0\colon\operatorname{Ho}(\mathscr{M})\rightarrow \operatorname{Ho}(\mathscr{N})$ were right adjoint to $\operatorname{Ho}(\mathscr{N})\subset \operatorname{Ho}(\mathscr{M})$? $\endgroup$ Apr 28, 2011 at 13:40
  • $\begingroup$ It is possible that, as suggested to me by Y. Laszlo, the commutativity $H^0\ holim=holim\ H^0$ still holds, not as a consequence of some formal argument, but of the so-called strictification theorem. I don't know this stuff, and hope some experts can explain. BTW, when the index category $I$ is infinite, how to show the existence of the holim? \\ @Denis-Charles: It's great to see you on MO! $\endgroup$
    – shenghao
    Apr 28, 2011 at 17:02
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    $\begingroup$ @Shengao. There is absolutely no chance that the formula $H^0 holim = holim H^0$ holds: to get a counter-example, you may restrict to dg categories with one objects (aka dg algebras); note that, for a diagram of dg algebras $A_i$, $holim A_i$ is computed as the homotopy limit of the underlying complexes. Therefore, such a formula would imply that any tower of algebras $A_{n+1}\to A_n$, $n\geq 0$, would satisfy the Mittag-Leffler condition... @Fernando. Hence $H^0$ cannot be a right adjoint. $\endgroup$ Apr 28, 2011 at 20:41
  • $\begingroup$ You're right :-( $\endgroup$ Apr 28, 2011 at 21:58

I'm sorry this answer is not on time. Actually for cosimplicial diagrams of dg-categories, which is mostly common in algebraic geometry, the homotopy limit is given by the dg-category of twisted complexes, under some conditions on the diagram. See this paper for details. (sorry for the self-citation.)


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