In mathematics, if A {\displaystyle A} is a subset of B , {\displaystyle B,} then the inclusion map is the function ι {\displaystyle \iota } that sends each element x {\displaystyle x} of A {\displaystyle A} to x , {\displaystyle x,} treated as an element of B : {\displaystyle B:} ι : A → B , ι ( x ) = x . {\displaystyle \iota :A\rightarrow B,\qquad \iota (x)=x.}
An inclusion map may also be referred to as an inclusion function, an insertion, or a canonical injection.
A "hooked arrow" (U+21AA ↪ RIGHTWARDS ARROW WITH HOOK) is sometimes used in place of the function arrow above to denote an inclusion map; thus: ι : A ↪ B . {\displaystyle \iota :A\hookrightarrow B.}
(However, some authors use this hooked arrow for any embedding.)
This and other analogous injective functions from substructures are sometimes called natural injections.
Given any morphism f {\displaystyle f} between objects X {\displaystyle X} and Y {\displaystyle Y} , if there is an inclusion map ι : A → X {\displaystyle \iota :A\to X} into the domain X {\displaystyle X} , then one can form the restriction f ∘ ι {\displaystyle f\circ \iota } of f . {\displaystyle f.} In many instances, one can also construct a canonical inclusion into the codomain R → Y {\displaystyle R\to Y} known as the range of f . {\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 ι ( x ⋆ y ) = ι ( x ) ⋆ ι ( y ) {\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 A {\displaystyle A} is a strong deformation retract of X , {\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 Spec ( R / I ) → Spec ( R ) {\displaystyle \operatorname {Spec} \left(R/I\right)\to \operatorname {Spec} (R)} and Spec ( R / I 2 ) → Spec ( R ) {\displaystyle \operatorname {Spec} \left(R/I^{2}\right)\to \operatorname {Spec} (R)} may be different morphisms, where R {\displaystyle R} is a commutative ring and I {\displaystyle I} is an ideal of R . {\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
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. 0-8218-1646-2 ↩
"Arrows – Unicode" (PDF). Unicode Consortium. Retrieved 2017-02-07. https://www.unicode.org/charts/PDF/U2190.pdf ↩
Chevalley, C. (1956). Fundamental Concepts of Algebra. New York, NY: Academic Press. p. 1. ISBN 0-12-172050-0. 0-12-172050-0 ↩