nLab
identity morphism
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
category theory
Concepts
Universal constructions
Theorems
Extensions
Applications
Contents
Idea
The identity morphism , or simply identity , of an object x x in some category C C is the morphism 1 x : x → x 1_x: x \to x , or id x : x → x \id_x: x \to x , which acts as a two-sided identity for composition .
Given a small category C C with set of objects C 0 C_0 and set of morphisms C 1 C_1 , the identity assigning function of C C is the function i : C 0 → C 1 i: C_0 \to C_1 that maps each object in C 0 C_0 to its identity morphism in C 1 C_1 .
For the generalisation to an internal category C C , see identity-assigning morphism .
In Set , the identity morphisms are the identity functions .
Last revised on December 1, 2019 at 08:18:08.
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