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 be a functor of dg-categories.
The functor is called fully faithful if for all objects the canonical morphisms of mapping complexes
are quasi-isomorphisms of chain complexes.
The functor is called essentially surjective if the induced functor on the homotopy categories is essentially surjective.
The functor 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.