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