nLab
identity element

See also identity type, unit, equivalence.

Identity elements

Definitions

Given an operation *:X×YY*\colon X \times Y \to Y, an element ee of XX is called a left identity for ** if e*a=ae * a = a for every element aa of YY. That is, the map YYY \to Y given by e*e * - is the identity function on YY.

If *:Y×XY*\colon Y \times X \to Y, then there is a similar concept of right identity.

If *:X×XX*\colon X \times X \to X, then ee is a 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 unit (although that term also has a broader meaning, and an operation that has an identity element is called unital or unitary. In particular, a magma whose underlying operation *:X×XX*\colon X \times X \to X has an identity is called a unital magma or a unitary magma.

Similarly, a 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 by Urs Schreiber (82.113.121.25)