A 1-morphism in a 2-category is an equivalence if there exists another 1-morphism the other way around, such that both are inverses up to invertible 2-morphisms.
In 2-category theory equivalences in a 2-category are sometimes also called “1-equivalences”, to distinguish these weakly invertible 1-morphisms from invertible 2-morphisms.
An equivalence in a 1-category regarded as a 2-category with only trivial 2-morphisms is just an isomorphism.
An equivalence in the 2-category Cat of all categories is an equivalence of categories.
An equivalence in the 2-category of algebras with bimodules as 1-morphisms and intertwiners as 2-morphisms (see here) is a Morita equivalence.
Last revised on May 5, 2017 at 17:06:16. See the history of this page for a list of all contributions to it.