nLab identity morphism

Idea

The identity morphism, or simply identity, of an object $x$ in some category $C$ is the morphism $1_x: x \to x$, or $\id_x: x \to x$, which acts as a two-sided identity for composition.

Given a small category $C$ with set of objects $C_0$ and set of morphisms $C_1$, the identity assigning function of $C$ is the function $i: C_0 \to C_1$ that maps each object in $C_0$ to its identity morphism in $C_1$.

For the generalisation to an internal category $C$, see identity-assigning morphism.

In Set, the identity morphisms are the identity functions.

Examples

Example

As morphisms in categories of topological vector spaces are also called linear operators (or just operators, for short), so the identity morphisms here are the identity operators.

Related concepts

• equality, equivalence

• identity function

• identity functor

• identity natural transformation

• identity type

• identity of indiscernibles