Inclusion map
In mathematics, if <math>A</math> is a subset of <math>B,</math> then the inclusion map is the function <math>\iota</math> that sends each element <math>x</math> of <math>A</math> to <math>x,</math> treated as an element of <math>B:</math> <math display=block>\iota : A\rightarrow B, \qquad \iota(x)=x.</math>
An inclusion map may also be referred to as an inclusion function, an insertion,<ref>Template:Cite book</ref> or a canonical injection.
A "hooked arrow" (Template:Unichar)<ref name="Unicode Arrows">{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> is sometimes used in place of the function arrow above to denote an inclusion map; thus: <math display=block>\iota: A\hookrightarrow B.</math>
(However, some authors use this hooked arrow for any embedding.)
This and other analogous injective functions<ref>Template:Cite book</ref> from substructures are sometimes called natural injections.
Given any morphism <math>f</math> between objects <math>X</math> and <math>Y</math>, if there is an inclusion map <math>\iota : A \to X</math> into the domain <math>X</math>, then one can form the restriction <math>f\circ \iota</math> of <math>f.</math> In many instances, one can also construct a canonical inclusion into the codomain <math>R \to Y</math> known as the range of <math>f.</math>
Applications of inclusion mapsEdit
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>\star,</math> to require that <math display=block>\iota(x\star y) = \iota(x) \star \iota(y)</math> is simply to say that <math>\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>A</math> is a strong deformation retract of <math>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 display=block>\operatorname{Spec}\left(R/I\right) \to \operatorname{Spec}(R)</math> and <math display=block>\operatorname{Spec}\left(R/I^2\right) \to \operatorname{Spec}(R)</math> may be different morphisms, where <math>R</math> is a commutative ring and <math>I</math> is an ideal of <math>R.</math>