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Although my experience with DG categories is pretty basic I find them to be a very neat tool for organizing (co-)homological techniques in algebraic geometry. For someone who has algebro-geometric application in mind they seem more attractive then stable $(\infty,1)-$categories which seem to carry data in a slightly more convoluted way (which I realize makes for more powerful techniques and generalizations).

So far though I've only seen a very limited amount of how actual algebraic geometry looks from a DG point of view and most of the stuff I read about DG categories was either definitions or general theory (papers by Toen for example). Here are several questions I have in mind:

  • What is the "correct" DG category associated to a scheme/algebraic space/stack?

  • Can the different possiblities here be organized as different "DG-stacks" (of certain dg-categories of sheaves) on the relevant site?

  • How can I see the classical "category" of derived categories as some kind of "category" of homotopy categories of dg-categories? (I'm putting category in brackets since i'm not sure that there's such an object, what I really want is to really understand the link between all the classical theory of derived categories and dg-categories). In particular the six functor formalism.

I realize that these question might not have a straight yes/no answer and so what I'm looking for is a kind of roadmap to the relevant litrature where the application and formalization of the place of dg-categories in algebraic geometry is discussed.

Main question: What are some relevant articles/notes/books which establish and discuss the details of the formalism of DG-categories in the algebro-geometric world?

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    $\begingroup$ Have you looked at Gaitsgory and Rozenblyum's new book? By the way, I disagree that stable categories "carry the data in a more convoluted way." After all, DG categories are just a model for (linear) stable categories, meaning the latter are the conceptual archetype for the former, although they came later historically. $\endgroup$ – Justin Campbell Jun 17 '16 at 15:54
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    $\begingroup$ @JustinCampbell Not yet I'll check it out. Just to clarify: the word "convoluted" wasn't meant to convey anything more than my own incompetence. Of which one crude example is that the definition of a DG category feels more approachable to me than that of a linear stable infinity category. $\endgroup$ – Saal Hardali Jun 17 '16 at 18:18
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    $\begingroup$ See Lunts-Orlov paper (arxiv.org/abs/0908.4187) on the uniqueness of DG-enhancements. $\endgroup$ – Sasha Jun 18 '16 at 21:20
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    $\begingroup$ One of the absolutely crucial papers on dg categories is To"en's arxiv.org/abs/math/0408337. An excellent pre-$\infty$-categorical overview is Keller's ICM address arxiv.org/abs/math/0601185 $\endgroup$ – David Ben-Zvi Jun 23 '16 at 3:11
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Let me try to address the bulleted questions and simultaneously advertise the G-R book everyone has mentioned. Since the main question was about literature, I could also mention Drinfeld's article "DG quotients of DG categories," which nicely summarizes the state of the general theory before $\infty$-categories shook everything up. However, it doesn't contain any algebraic geometry.

If $X = \text{Spec } A$ is an affine scheme, it's reasonable to define the category of quasicoherent sheaves $\text{QCoh}(X) := A\text{-mod}$ as the category of $A$-modules. Any other definition (e.g. via Zariski sheaves) must reproduce this answer anyway. If we understand this as the derived category of $A$-modules, then there is a canonical DG model: the homotopically projective complexes in the sense of Drinfeld's article.

The next step is to construct $\text{QCoh}(X)$ for $X$ not necessarily affine. So write $X = \cup_i \text{Spec } A_i$ as a union of open affines (say $X$ is separated to simplify things). It would be great if we could just "glue" the categories $A_i\text{-mod}$, the way that we compute global sections of a sheaf as a certain equalizer. Concretely, a complex of sheaves on $X$ should consist of complexes of $A_i$-modules for all $i$, identified on overlaps via isomorphisms satisfying cocycle "conditions" (really extra data). This is the kind of thing that totally fails in the triangulated world: limits of 1-categories just don't do the trick. Even if we work with the DG enhancements, DG categories do not form a DG category, so this doesn't help.

As you might have guessed, this is where $\infty$-categories come to the rescue. Let me gloss over details and just say that there is a (stable, $k$-linear) $\infty$-category attached to a DG category such as $A$-mod, called its DG nerve. If we take the aforementioned equalizer in the $\infty$-category of $\infty$-categories, then we do get the correct $\infty$-category $\text{QCoh}(X)$, in the sense that its homotopy category is the usual derived category of quasicoherent sheaves on $X$. (Edit: As Rune Haugseng explains in the comments, it's actually necessary to take the limit of the diagram of $\infty$-categories you get by applying $\text{QCoh}$ to the Cech nerve of the covering. The equalizer is a truncated version of this.)

But, you might be thinking, I could have just constructed a DG model for $\text{QCoh}(X)$ using injective complexes of Zariski sheaves or something. That's true, and obviously suffices for tons of applications, but as soon as you want to work with more general objects than schemes you're hosed. True, there are workarounds using DG categories for Artin stacks, but the theory gets very technical very fast.

If we instead accept the inevitability of $\infty$-categories, we can make the following bold construction. A prestack is an arbitrary functor from affine schemes to $\infty$-groupoids (i.e. spaces in the sense of homotopy theory). For example, affine schemes are representable prestacks, but prestacks also include arbitrary schemes and Artin stacks. Then for any prestack $\mathscr{X}$ we can define $\text{QCoh}(\mathscr{X})$ to be the limit of the $\infty$-categories $A\text{-mod}$ over the $\infty$-category of affine schemes $\text{Spec } A$ mapping to $\mathscr{X}$. A cofinality argument for Zariski atlases shows this agrees with our previous definition for $\mathscr{X}$ a scheme.

For example, if $\mathscr{X} = \text{pt}/G$ is the classifying stack of an algebraic group $G$, then the homotopy category of $\text{QCoh}(\mathscr{X})$ is the derived category of representations of $G$. Even cooler: if $X$ is a scheme the de Rham prestack $X_{\text{dR}}$ is defined by $$\text{Map}(S,X_{\text{dR}}) := \text{Map}(S_{\text{red}},X).$$ Then, at least if $k$ has characteristic zero, our definition of $\text{QCoh}(X_{\text{dR}})$ recovers the derived category of crystals on $X$, which can be identified with $\mathscr{D}$-modules. So we put two different ``flavors" of sheaf theory on an equal footing.

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    $\begingroup$ Great answer! slight quibble: I believe sheaves on the de Rham prestack in positive characteristic don't match what arithmetic geometers usually mean by crystals (i.e., on the crystalline rather than infinitesimal site) - for that you'd need a divided power analog of the de Rham functor. $\endgroup$ – David Ben-Zvi Jun 23 '16 at 3:13
  • $\begingroup$ Thanks David, I fixed it. A divided power analogue of the de Rham prestack sounds cool, has anyone written about that? $\endgroup$ – Justin Campbell Jun 23 '16 at 15:15
  • $\begingroup$ As far as I know nothing is written in that language, though wouldn't be surprised if it's somewhere e.g. in Bhatt's papers. $\endgroup$ – David Ben-Zvi Jun 23 '16 at 15:25
  • $\begingroup$ Very naively you'd define it as the quotient of $X$ by the PD completion of $X\times X$ along the diagonal, rather than the formal neighborhood of the diagonal... $\endgroup$ – David Ben-Zvi Jun 23 '16 at 15:30
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    $\begingroup$ This might be overly pedantic, but to get the right $\infty$-category of quasicoherent sheaves I think you need to take the limit of the cosimplicial diagram that takes all the iterated intersections into account, rather than just an equalizer. (This is a general theme in passing from 1-categories to $\infty$-categories. In a sense global sections are "really" always a cosimplicial limit like this, but in a 1-category that's equivalent to an equalizer - and in a 2-category you only have to take triple intersections into account, etc.) $\endgroup$ – Rune Haugseng Jun 23 '16 at 22:48
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There are plenty of interesting dg-categories one can associate to a scheme. From the point of view of six functor yoga, these should be viewed as "categories of coefficients" for cohomology theories. For example, the derived category of quasi-coherent sheaves (or its various variants) is the category of coefficients for coherent cohomology, just as the derived category of $\ell$-adic sheaves is the category of coefficients for $\ell$-adic cohomology, or motivic complexes are the coefficients for motivic cohomology.

These have been studied for decades using the language of triangulated categories, but it is well-known that they can each be defined as dg-categories. As the base varies, they form stacks of dg-categories with good descent properties (depending on the category of coefficients). In fact, in each of these examples, there is much more structure: there is a whole six functor formalism, categorifying the standard features of cohomology theories, like Künneth formulas, Poincaré duality, Gysin maps, etc. The six functor formalisms also lift to the dg-level.

For any given category of coefficients you might be interested in, there are certainly plenty of references, though most of them will be written in the language of triangulated categories. You shouldn't be bothered by this: there is a very large amount of interesting algebraic geometry you can do in this language, and anyway the arguments can be translated to more modern language without changing very much how they look.

On the other hand, if you are specifically interested in seeing the power of the modern language, one important point is the failure of descent at the triangulated level. The book of Gaitsgory-Rozenblyum is a great place to see descent arguments in practice. Another very good reference is the work of Bhatt-Scholze.

I should note that both these references actually use the language of $(\infty,1)$-categories instead of dg-categories. In fact, Gaitsgory-Rozenblyum's definition of dg-category is just a $k$-linear stable $(\infty,1)$-category. If you work in the $(\infty,1)$-category of dg-categories, you don't see any difference between this and the classical definition. If you do want to work more set-theoretically, you will have to pay for the psychological comfort by doing a lot of extra work to make sure the constructions you do are homotopically meaningful (e.g. whenever you take a tensor product or internal hom). If that's your preference, then I would suggest papers of Kuznetsov, Lunts and Orlov (which definitely contain a lot of interesting and beautiful mathematics).

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Besides Gaitsgory-Rozenblyum (http://www.math.harvard.edu/~gaitsgde/GL/), you might try looking at Lee Cohn's work (http://arxiv.org/abs/1308.2587), which establishes some equivalence between the approaches. Abstract follows:

Differential Graded Categories are k-linear Stable Infinity Categories

Lee Cohn (Submitted on 12 Aug 2013) We describe a comparison between pretriangulated differential graded categories and certain stable infinity categories. Specifically, we use a model category structure on differential graded categories over k (a field of characteristic 0) where the weak equivalences are the Morita equivalences, and where the fibrant objects are in particular pretriangulated differential graded categories. We show the underlying infinity category of this model category is equivalent to the infinity category of k-linear stable infinity categories.

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