Inclusion map

In mathematics, if ${\displaystyle A}$ is a subset of ${\displaystyle B,}$ then the inclusion map is the function ${\displaystyle \iota }$ that sends each element ${\displaystyle x}$ of ${\displaystyle A}$ to ${\displaystyle x,}$ treated as an element of ${\displaystyle B:}$

An inclusion map may also referred to as an inclusion function, an insertion,^[1] or a canonical injection.

A "hooked arrow" (U+21AA ↪ RIGHTWARDS ARROW WITH HOOK)^[2] is sometimes used in place of the function arrow above to denote an inclusion map; thus:

(However, some authors use this hooked arrow for any embedding.)

This and other analogous

are sometimes called natural injections.

Given any morphism ${\displaystyle f}$ between

${\displaystyle X}$ and ${\displaystyle Y}$, if there is an inclusion map ${\displaystyle \iota :A\to X}$ into the domain ${\displaystyle X}$, then one can form the restriction ${\displaystyle f\circ \iota }$ of ${\displaystyle f.}$ In many instances, one can also construct a canonical inclusion into the codomain ${\displaystyle R\to Y}$ known as the range of ${\displaystyle f.}$

Applications of inclusion maps

Inclusion maps tend to be homomorphisms of algebraic structures; thus, such inclusion maps are embeddings. More precisely, given a substructure closed under some operations, the inclusion map will be an embedding for tautological reasons. For example, for some binary operation ${\displaystyle \star ,}$ to require that

is simply to say that ${\displaystyle \star }$ is consistently computed in the sub-structure and the large structure. The case of a means such constants must already be given in the substructure.

Inclusion maps are seen in algebraic topology where if ${\displaystyle A}$ is a

of ${\displaystyle X,}$ the inclusion map yields an ).

Inclusion maps in

affine schemes

, for which the inclusions

$${\displaystyle \operatorname {Spec} \left(R/I\right)\to \operatorname {Spec} (R)}$$

and

$${\displaystyle \operatorname {Spec} \left(R/I^{2}\right)\to \operatorname {Spec} (R)}$$

may be different morphisms, where ${\displaystyle R}$ is a commutative ring and ${\displaystyle I}$ is an ideal of ${\displaystyle R.}$

See also

• Cofibration – continuous mapping between topological spacesPages displaying wikidata descriptions as a fallback
• Identity function – In mathematics, a function that always returns the same value that was used as its argument

References