# Inclusion map

In mathematics, if is a subset of , then the **inclusion map** (also **inclusion function**, **insertion**,^{[1]} or **canonical injection**) is the function that sends each element, , of to , treated as an element of :

A "hooked arrow" 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 , then one can form the restriction *fi* of *f*. In many instances, one can also construct a canonical inclusion into the codomain *R*→*Y* 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 , to require that

is simply to say that 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

and

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

## See also

## Notes

- ↑ Mac Lane, S.; Birkhoff, G. (1967),
*Algebra*, page 5, says "Note that “insertion” is a function and “inclusion” a relation ; every inclusion relation gives rise to an insertion function." - ↑ Chevalley, C. (1956),
*Fundamental Concepts of Algebra*, page 1 - ↑ I.e., objects that have pullbacks; these are called covariant in an older and unrelated terminology

## References

- Chevalley, C. (1956),
*Fundamental Concepts of Algebra*, Academic Press, New York, ISBN 0-12-172050-0. - Mac Lane, S.; Birkhoff, G. (1967),
*Algebra*, AMS Chelsea Publishing, Providence, Rhode Island, ISBN 0-8218-1646-2.