nLab equivalence of dg-categories

Idea

An equivalence of dg-categories is in an equivalence in the (infinity,1)-category of dg-categories. They are analogous to Dwyer-Kan equivalences of simplicial categories.

Definition

Let $u : C \to D$ be a functor of dg-categories.

Definition

The functor $u$ is called fully faithful if for all objects $x,y \in C$ the canonical morphisms of mapping complexes

$$Map(x,y) \longrightarrow Map(u(x), u(y))$$

are quasi-isomorphisms of chain complexes.

The functor $u$ is called essentially surjective if the induced functor on the homotopy categories is essentially surjective.

The functor $u$ is called an equivalence or Dwyer-Kan equivalence if it is fully faithful and essentially surjective.

In the literature the term quasi-equivalence is often used for this notion.

Equivalences are dg-categories are precisely the equivalences in the (infinity,1)-category of dg-categories, which is presented by the Dwyer-Kan model structure on dg-categories.

References

See the references at dg-category.