Inclusion map

then the inclusion map is the function

that sends each element

treated as an element of

An inclusion map may also be referred to as an inclusion function, an insertion,[1] or a canonical injection.

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.

, then one can form the restriction

In many instances, one can also construct a canonical inclusion into the codomain

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.

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

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