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" (
U+21AA ↪ RIGHTWARDS ARROW WITH HOOK)^{
[2]} is sometimes used in place of the function arrow above to denote an inclusion map; thus:

(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 between objects and , if there is an inclusion map into the domain then one can form the restriction of In many instances, one can also construct a canonical inclusion into the codomain known as the range of

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

Inclusion maps are seen in algebraic topology where if is a strong deformation retract of 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 is a
commutative ring and is an
ideal of

- Cofibration
- Identity function – In mathematics, a function that always returns the same value that was used as its argument

**^**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.