Derived category of coherent sheaves as modules over dg-category

Question

I'm trying to learn dg-categories and got stuck with the motivation for considering modules over dg-categories. By definition, for a dg-category $A$ the category of modules over it is $$Mod{-}A=Fun\left(A^{op}, C(mod{-}k)\right),$$ that is the dg-category of contravariant dg-functors from $A$ to the dg-category of complexes of $k$-modules, and its homotopy category is denoted as $D(A)=H^0(Mod{-}A)$. So what should I take as $A$ to get the derived category $D_b(Coh(X))$ of coherent sheaves on an algebraic variety as $D(A)$?

Answer 1

Summary: Describing the bounded derived category as modules over a dg-category is a non-canonical two-step process. In the first step, you realize the category in question as the homotopy category of some dg-category (not of modules over it!), and in the second step, you note that there's no difference between the just obtained dg-category and the so-called perfect modules over it.

In summary, you therefore get a description of ${\mathbf D}^b(\text{Coh}(X))$ as the perfect modules over some dg-category.

Note: You cannot hope for a description as a description as all modules over some dg-category, because this would either be trivial or essentially big, whereas ${\mathbf D}^b(\text{Coh}(X))$ is essentially small.

First step: As you probably know, ${\mathbf D}^b(\text{Coh}(X))$ is equivalent to the homotopy category ${\mathbf K}^{+,b,\text{Coh}}(\text{QCoh-Inj}(X))$ of bounded below complexes of injective quasi-coherent sheaves with bounded and coherent cohomology. This in turn can be canonically realized as the homotopy category of the dg-category ${\mathbf C}^{+,b,\text{Coh}}(\text{QCoh-Inj}(X))$ which is defined alike, with homomorphism-complexes as the dg-homs.

Second step: The dg-category ${\mathbf C}:={\mathbf C}^{+,b,\text{Coh}}(\text{QCoh-Inj}(X))$ is pretriangulated and ${\mathbf D}^b(\text{Coh}(X))$ is idempotent complete, so the Yoneda embedding $\textbf{C}\to \textbf{C}\text{-Mod}$ induces a fully faithful functor $\text{Ho}(\textbf{C})\to\text{Ho}(\textbf{C}\text{-Mod})$ identifying $\text{Ho}(\textbf{C})$ with the full subcategory of perfect dg-modules over $\textbf{C}$.