Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
identity type, equivalence of types, definitional isomorphism
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
A 1-morphism in a 2-category is an equivalence if there exists another 1-morphism the other way around, such that the two are inverses of each other up to invertible 2-morphisms.
A 1-morphism in a 2-category is called an equivalence if there exists a 1-morphism and invertible 2-morphisms , .
Alternatively, some use equivalence to denote the whole 4-tuple
Then being an equivalence is an extra structure put on a 1-morphism in a 2-category, not merely an extra property.
Such a 4-tuple is called an adjoint equivalence if it obeys the zigzag identities, and in that case (and only in that case) is called the unit and is called the counit. Given an equivalence in a 2-category, it can always be ‘improved’ to become an adjoint equivalence, simply by redefining or .
Equivalences in a 2-category are sometimes also called “1-equivalences”, to distinguish them from invertible 2-morphisms, which are also called “2-isomorphisms”.
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 November 13, 2024 at 07:46:47. See the history of this page for a list of all contributions to it.