# Inclusion map

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

In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, or canonical injection) is the function i that sends each element, x of A to x, treated as an element of B:

$i: A\rightarrow B, \qquad i(x)=x.$

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

This and other analogous injective functions from substructures are sometimes called natural injections.

Given any morphism between objects X and Y, if there is an inclusion map into the domain $i : A\rightarrow X$, 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, given the very definition by restriction of what one checks. For example, for a binary operation $\star$, to require that

$i(x\star y)=i(x)\star i(y)$

is simply to say that $\star$ 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 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)

and

Spec(R/I2)Spec(R)

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

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