Inclusion map

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[math]\displaystyle{ A }[/math] is a subset of [math]\displaystyle{ B, }[/math] and [math]\displaystyle{ B }[/math] is a superset of [math]\displaystyle{ A. }[/math]

In mathematics, if [math]\displaystyle{ A }[/math] is a subset of [math]\displaystyle{ B, }[/math] then the inclusion map (also inclusion function, insertion,[1] or canonical injection) is the function [math]\displaystyle{ \iota }[/math] that sends each element [math]\displaystyle{ x }[/math] of [math]\displaystyle{ A }[/math] to [math]\displaystyle{ x, }[/math] treated as an element of [math]\displaystyle{ B: }[/math] [math]\displaystyle{ \iota : A\rightarrow B, \qquad \iota(x)=x. }[/math]

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: [math]\displaystyle{ \iota: A\hookrightarrow B. }[/math]

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

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 [math]\displaystyle{ \star, }[/math] to require that [math]\displaystyle{ \iota(x\star y) = \iota(x) \star \iota(y) }[/math] is simply to say that [math]\displaystyle{ \star }[/math] 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 [math]\displaystyle{ A }[/math] is a strong deformation retract of [math]\displaystyle{ X, }[/math] 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 [math]\displaystyle{ \operatorname{Spec}\left(R/I\right) \to \operatorname{Spec}(R) }[/math] and [math]\displaystyle{ \operatorname{Spec}\left(R/I^2\right) \to \operatorname{Spec}(R) }[/math] may be different morphisms, where [math]\displaystyle{ R }[/math] is a commutative ring and [math]\displaystyle{ R }[/math] is an ideal of [math]\displaystyle{ R. }[/math]

See also


  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 SU and "inclusion" a relation SU; every inclusion relation gives rise to an insertion function." 
  2. "Arrows – Unicode". Unicode Consortium. 
  3. Chevalley, C. (1956). Fundamental Concepts of Algebra. New York, NY: Academic Press. p. 1. ISBN 0-12-172050-0.