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:CDu : C \to D be a functor of dg-categories.

Definition

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

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

are quasi-isomorphisms of chain complexes.

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

The functor uu 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.

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