nLab derived dg-category of a dg-category

Idea

The derived dg-category of a dg-category is the dg-localization of the dg-category of dg-modules at the class of quasi-isomorphisms. The result is a pretriangulated dg-category. However an important caveat is that, even if the original dg-category is pretriangulated, it is strictly contained inside its derived dg-category.

Despite the terminology, the derived dg-category is not quite analogous to the derived category of an abelian category. Rather it should be thought of as the analogue of the simplicially enriched category of simplicial presheaves, hence a presentation of the (infinity,1)-category of (infinity,1)-presheaves.

Definition

Let $T$ be a dg-category.

Definition

The derived dg-category of $T$, denoted $D(T)$, is the dg-localization of the category of dg-modules over $T$ at the class of objectwise quasi-isomorphisms:

$$D(T) = dg-mod_T[qis^{-1}].$$

$D(T)$ can also be described as the dg-category presented by the projective dg-model structure on dg-modules.

Properties

Lemma

The dg-category $D(T)$ is pretriangulated.

Lemma

There exists a fully faithful functor of dg-categories

$$T \hookrightarrow D(T)$$

called the dg-Yoneda embedding.

Related concepts

• dg-module
• dg-Yoneda embedding

References

• B. Toen, The homotopy theory of dg-categories and derived Morita theory, arXiv:math/0408337.