nLab
natural equivalence
Contents
Context
Equality and Equivalence
equivalence

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

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

2-Category theory
2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

Higher category theory
higher category theory

Basic concepts
Basic theorems
Applications
Models
Morphisms
Functors
Universal constructions
Extra properties and structure
1-categorical presentations
Contents
Definition
Often, by a natural equivalence is meant specifically an equivalence in a 2-category of 2-functors .

But more generally it is an equivalence between any kind of functors in higher category theory :

The components of a natural equivalence are equivalences between the objects in the codomain of the functors. This is what the term “natural equivalence” refers to: its a collection of equivalences between objects which are compatible (“natural”) with the morphisms between these objects, and higher morphisms between those.

Last revised on April 22, 2021 at 23:49:51.
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