nLab
identity natural transformation
Contents
Context
Equality and Equivalence
equivalence

equality (definitional , propositional , computational , judgemental , extensional , intensional , decidable )

identity type , equivalence in homotopy type theory

isomorphism , weak equivalence , homotopy equivalence , weak homotopy equivalence , equivalence in an (∞,1)-category

natural equivalence , natural isomorphism

gauge equivalence

Examples.

principle of equivalence

equation

fiber product , pullback

homotopy pullback

Examples.

linear equation , differential equation , ordinary differential equation , critical locus

Euler-Lagrange equation , Einstein equation , wave equation

Schrödinger equation , Knizhnik-Zamolodchikov equation , Maurer-Cartan equation , quantum master equation , Euler-Arnold equation , Fuchsian equation , Fokker-Planck equation , Lax equation

Category theory
Contents
Definition
The identity natural transformation on a functor $F: C \to D$ is the natural transformation $id_F: F \to F$ that maps each object $x$ of $C$ to the identity morphism $id_{F(x)}$ in $D$ .

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

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

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

Last revised on August 1, 2022 at 19:27:11.
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