, identity type , unit . equivalence
Equality and 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
principle of equivalence
fiber product, pullback
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 Identity elements
Given an operation
, an element * : X × Y → Y *\colon X \times Y \to Y of e e is called a X X left identity for if * * for every element e * a = a e * a = a of a a . That is, the map Y Y given by Y → Y Y \to Y is the e * − e * - identity function on . Y Y
, then there is a similar concept of * : Y × X → Y *\colon Y \times X \to Y right identity.
, then * : X × X → X *\colon X \times X \to X is a e e two-sided identity, or simply identity, if it is both a left and right identity.
Historically, identity elements (as above) came first, then
identity functions, and then identity morphisms. These are all the same basic idea, however: an identity morphism is an identity element for the operation of composition.
An identity is sometimes called a
(although that term also has a broader meaning, and an operation that has an identity element is called unit unital or unitary. In particular, a magma whose underlying operation has an identity is called a * : X × X → X *\colon X \times X \to X unital magma or a unitary magma.
unit law is the statement that a given operation has an identity element. In higher category theory, we generalise from the property of uniticity/unitality to the structure of a unitor.
Revised on November 1, 2013 12:35:18