nLab
identity function
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

EulerLagrange equation, Einstein equation, wave equation

Schrödinger equation, KnizhnikZamolodchikov equation, MaurerCartan equation, quantum master equation, EulerArnold equation, Fuchsian equation, FokkerPlanck equation, Lax equation
Contents
Definition
Given a set $S$, the identity function on $S$ is the function $id_S\colon S \to S$ that maps any element $x$ of $S$ to itself:
$id_S = (x \mapsto x) = \lambda x.\; x ;$
or equivalently,
$id_S(x) = x .$
The identity functions are the identity morphisms in the category Set of sets.
More generally, in any concrete category, the identity morphism of each object is given by the identity function on its underlying set.
Revised on July 2, 2017 09:25:23
by
Urs Schreiber
(88.77.226.246)