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

1 Answer

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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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 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 :) – Dylan Wilson Nov 24 at 6:20
Another point is that I think the proof of (3) would basically reduce to showing that the $\infty$-category of k-module spectra is the same as the $\infty$-category of chain complexes over $k$. This is probably in the literature in the language of model categories, maybe even in Shipley's paper "HZ-algebra spectra are differential graded algebras". Maybe in general it's true that R-linear spectra are the same as DG R-modules when R is a discrete ring? I haven't read the paper so I don't know... – Dylan Wilson Nov 24 at 6:24
... so maybe the characteristic zero part only enters the picture when we want to relate this to the $A_\infty$ story. But I really don't know about this stuff, so can someone else fill in here? I'm interested! – Dylan Wilson Nov 24 at 6:24
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 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". – Eric Wofsey Nov 24 at 8:14
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 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. – Akhil Mathew Nov 24 at 16:12
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