Inclusion map

In mathematics, if $A$ is a subset of $B$ , then the inclusion map (also inclusion function, insertion,^[1] or canonical injection) is the function $ι$ that sends each element $x$ of $A$ to $x$ , treated as an element of $B$ :

\iota :A\rightarrow B,\qquad \iota (x)=x.

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:

\iota :A\hookrightarrow B.

(On the other hand, this notation is sometimes reserved for embeddings.)

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

Given any morphism $f$ between objects $X$ and $Y$ , if there is an inclusion map into the domain $ι : A \to X$ , then one can form the restriction $f ι$ of $f$ . In many instances, one can also construct a canonical inclusion into the codomain $R \to Y$ known as the range of $f$ .

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 $⋆$ , to require that

\iota (x\star y)=\iota (x)\star \iota (y)

is simply to say that $⋆$ 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 $A$ is a strong deformation retract of $X$ , 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

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

and

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

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

References

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

[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 $S \to U$ and "inclusion" a relation $S \subset U$ ; every inclusion relation gives rise to an insertion function.

[Unicode_Arrows-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.

[1]

[2]

[3]

Applications of inclusion maps

See also

References