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What is the relation between dg-categories and stable $\infty$-categories?

Given a dg-category one can form its dg-nerve and get a $\infty$-category (which will be stable if the dg-category is?). Can one turn a stable $\infty$-category into a dg-category or $A_\infty$-category somehow?

I have heard the statement that at least over a field of characteristic zero the theories of stable $\infty$-categories and dg-categories are "equivalent".

What would be a precise formulation of this statement and what would be a reference?

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    $\begingroup$ What's a stable $\infty$-category over a field (of characteristic zero)? $\endgroup$
    – Fernando Muro
    Commented Nov 23, 2012 at 15:40

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Here are a few observations...

  1. I think there exist stable infinity categories that are not the dg-nerve (resp. $A_\infty$-nerve) of a dg-category (resp. $A_\infty$ category). In particular, the category of spectra should not arise in this way. I think Keller has a paper on differential graded categories that answers this question; he notes at some point that the homotopy category of spectra is not "algebraic" but that homotopy categories of differential graded categories are (and in fact sort of encompass all such algebraic categories.) Basically it comes down to something like the existence of Hopf maps. Now- could one define somehow the "closest dg-category approximation" to a given stable infty category? Probably. I don't know how. Or maybe I could come up with how, but I'm not sure how useful this would be if the functor wasn't an equivalence?
  2. To answer Fernando's question, see DAG X.5 or DAG VII.6.2. That is, a stable $\infty$-category over a field $k$ is a presentable, stable $\infty$-category "equipped with an action of the monoidal $\infty$-category of $k$-module spectra". Unless I'm mistaken I think this basically implies that it is enriched and tensored over k-module spectra.
  3. Here would be a precise formulation of the statement about categories over a field of characteristic zero: The dg-nerve functor induces an equivalence of $\infty$-categories between the $\infty$-category underlying the model category of dg-categories over k and the $\infty$-category of stable, k-linear $\infty$-categories. (I don't mean to overwhelm with "infinities", I would state this in the perhaps friendlier world of model categories, but I'm not sure what precisely the model category is that corresponds to stable, k-linear $\infty$-categories.) I don't know of a reference for a proof, though Lurie alludes to this a lot. It would be great if someone wrote this down!
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    $\begingroup$ I should mention that the "Hopf maps" proof that the homotopy category of spectra does not arise from algebraic procedures is due, I believe, to Fernando Muro :) $\endgroup$
    – Dylan Wilson
    Commented Nov 24, 2012 at 6:20
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    $\begingroup$ HR-linear spectra are the same as dg R-modules for any R. What's special about characteristic $0$ is that if the homotopy groups of a spectrum are $\mathbb{Q}$-modules, that automatically implies that the spectrum is an $H{\mathbb Q}$-module (that is to say, the localization of the sphere spectrum $S_{\mathbb Q}$ is the same as $H{\mathbb Q}$). Thus if you have a stable $\infty$-category in which the groups of maps between objects are always ${\mathbb Q}$-modules, it automatically is "dg" or "algebraic". $\endgroup$
    – Eric Wofsey
    Commented Nov 24, 2012 at 8:14
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    $\begingroup$ Hey Dylan -- ch. 7 of "Higher Algebra" has a really clean argument that for $R$ a discrete $E_1$-ring, the $\infty$-category of $R$-module spectra is equivalent to the derived $\infty$-category of $\pi_0 R$-modules. The idea is that the derived $\infty$-category always has a universal property, so that it's easy to produce a functor in one direction. Then you can check that it's an equivalence by playing with the $t$-structure on both sides. $\endgroup$
    – Akhil Mathew
    Commented Nov 24, 2012 at 16:08
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    $\begingroup$ Also, in response to Jan's point: a dg-category is basically a category enriched in chain complexes ($H \mathbb{Z}$-module spectra). However, that doesn't mean that the associated $\infty$-category is actually stable. You probably want to say that pull-back squares are push-outs, or something equivalent to that. $\endgroup$
    – Akhil Mathew
    Commented Nov 24, 2012 at 16:12
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    $\begingroup$ I believe the relevant "stability" condition for dg categories is that they be pre-triangulated. OTOH it depends how you ask the question - in the appropriate (Morita?) model structure I think all dg categories will be weakly equivalent to pre-triangulated ones anyway. $\endgroup$
    – David Ben-Zvi
    Commented Nov 24, 2012 at 21:03
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See the recent paper

  • Lee Cohn, Differential graded categories are k-linear stable infinity categories, arXiv:1308.2587

where a proof has been written down. The precise statement is that the underlying $(\infty,1)$-category associated to the Morita model structure on dg-categories over $k$ (where fibrant objects are karoubian pretriangulated dg-categories) is equivalent to the $(\infty,1)$-category of karoubian stable $k$-linear $(\infty,1)$-categories.

Update: Though Cohn works over characteristic zero, Bertrand Toen told me his arguments in fact hold in arbitrary characteristic.

Also, see the new paper

  • Giovanni Faonte, Simplicial nerve of an A-infinity category, arXiv:1312.2127

where it is proved that the dg-nerve functor takes pretriangulated dg-categories to stable $(\infty,1)$-categories.

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