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 ${\displaystyle B:}$

${\displaystyle A}$ is a subset of ${\displaystyle B,}$ and ${\displaystyle B}$ is a superset of ${\displaystyle A.}$

$${\displaystyle \iota :A\rightarrow B,\qquad \iota (x)=x.}$$

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:

$${\displaystyle \iota :A\hookrightarrow B.}$$

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

This and other analogous injective functions^[3] from substructures are sometimes called natural injections.

Given any morphism ${\displaystyle f}$ between objects ${\displaystyle X}$ 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

$${\displaystyle \iota (x\star y)=\iota (x)\star \iota (y)}$$

 

is simply to say that ${\displaystyle \star }$  is consistently computed in the sub-structure and the large structure. The case of a unary operation is similar; but one should also look at nullary operations, which pick out a constant element. Here the point is that closure means such constants must already be given in the substructure.

Inclusion maps are seen in algebraic topology where if ${\displaystyle A}$  is a strong deformation retract of ${\displaystyle X,}$  the inclusion map yields an isomorphism between all homotopy groups (that is, it is a homotopy equivalence).

Inclusion maps in geometry come in different kinds: for example embeddings of submanifolds. Contravariant objects (which is to say, objects that have pullbacks; these are called covariant in an older and unrelated terminology) such as differential forms restrict to submanifolds, giving a mapping in the other direction. Another example, more sophisticated, is that of 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

1. MacLane, S.; Birkhoff, G. (1967). Algebra. Providence, RI: AMS Chelsea Publishing. p. 5. ISBN 0-8218-1646-2. Note that "insertion" is a function S → U and "inclusion" a relation S ⊂ U; every inclusion relation gives rise to an insertion function.
2. "Arrows – Unicode" (PDF). Unicode Consortium. Retrieved 2017-02-07.
3. Chevalley, C. (1956). Fundamental Concepts of Algebra. New York, NY: Academic Press. p. 1. ISBN 0-12-172050-0.