nLab equivalence in a 2-category

Idea

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.

Definition

A 1-morphism $f: x \to y$ in a 2-category is called an equivalence if there exists a 1-morphism $g: y \to x$ and invertible 2-morphisms $\eta : 1_y \xrightarrow{\sim} f g$, $\epsilon: g f \xrightarrow{\sim} 1_x$.

Alternatively, some use equivalence to denote the whole 4-tuple

$$(f: x \to y, g: y \to x, \eta : 1_y \xrightarrow{\sim} f g, \epsilon: g f \xrightarrow{\sim} 1_x)$$

Then being an equivalence is an extra structure put on a 1-morphism $f: x \to y$ 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) $\eta$ is called the unit and $\epsilon$ is called the counit. Given an equivalence in a 2-category, it can always be ‘improved’ to become an adjoint equivalence, simply by redefining $\epsilon$ or $\eta$.

Equivalences in a 2-category are sometimes also called “1-equivalences”, to distinguish them from invertible 2-morphisms, which are also called “2-isomorphisms”.

Examples

• 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.

Related concepts

• homotopy equivalence

• equivalence in an (infinity,1)-category