nLab identity natural transformation

Contents

Context

Equality and Equivalence

Category theory

Contents

Definition

The identity natural transformation on a functor F:CDF: C \to D is the natural transformation id F:FFid_F: F \to F that maps each object xx of CC to the identity morphism id F(x)id_{F(x)} in DD.

The identity natural transformations are themselves the identity morphisms for vertical composition of natural transformations in the functor category D CD^C and in the 2-category Cat.

One must be aware that when we say that a natural transformation α A:F(A)G(A)\alpha_{A}:F(A) \rightarrow G(A) is the identity, it doesn’t mean only that F(A)=G(A)F(A)=G(A) for every object AA but also that F(f)=G(f)F(f)=G(f) for every morphism ff ie. that the two functors FF and GG are equal.

Not taking care of this can lead to redundant definitions as for permutative categories.

Last revised on August 1, 2022 at 15:27:11. See the history of this page for a list of all contributions to it.