# Inclusion map

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A is a subset of B, and B is a superset of A.

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

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

A "hooked arrow" ${\displaystyle \hookrightarrow }$ is sometimes used in place of the function arrow above to denote an inclusion map.

This and other analogous injective functions[2] 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 ${\displaystyle \iota :A\rightarrow X}$, then one can form the restriction fi of f. In many instances, one can also construct a canonical inclusion into the codomain RY 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 sub-structure closed under some operations, the inclusion map will be an embedding for tautological reasons. For example, for a 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 A is a strong deformation retract of X, the inclusion map yields an isomorphism between all homotopy groups (i.e. is a homotopy equivalence).

Inclusion maps in geometry come in different kinds: for example embeddings of submanifolds. Contravariant objects[3] 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

${\displaystyle \operatorname {Spec} (R/I)\to \operatorname {Spec} (R)}$

and

${\displaystyle \operatorname {Spec} (R/I^{2})\to \operatorname {Spec} (R)}$

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

1. ^ Mac Lane, S.; Birkhoff, G. (1967), Algebra, page 5, says "Note that “insertion” is a function ${\displaystyle S\to U}$ and “inclusion” a relation ${\displaystyle S\subset U}$; every inclusion relation gives rise to an insertion function."