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

An inclusion map may also be 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 ${\displaystyle f.}$

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 ${\displaystyle R.}$

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