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.

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

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.

See the references at dg-category.

Created on January 7, 2015 at 12:30:24. See the history of this page for a list of all contributions to it.