# Inclusion map

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

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:

(On the other hand, this notation is sometimes reserved for embeddings.)

This and other analogous injective functions[3] 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 *ι* : *A* → *X*, then one can form the restriction f ι 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 substructure closed under some operations, the inclusion map will be an embedding for tautological reasons. For example, for some 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 (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

and

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

## See also

## References

- MacLane, S.; Birkhoff, G. (1967).
*Algebra*. Providence, RI: AMS Chelsea Publishing. p. 5. ISBN 0-8218-1646-2.Note that “insertion” is a function

*S*→*U*and "inclusion" a relation*S*⊂*U*; every inclusion relation gives rise to an insertion function. - "Arrows – Unicode" (PDF). Unicode Consortium. Retrieved 2017-02-07.
- Chevalley, C. (1956).
*Fundamental Concepts of Algebra*. New York, NY: Academic Press. p. 1. ISBN 0-12-172050-0.