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:



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> * **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.**
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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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DG categories in algebraic geometry - guide to the literature?
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.


