Inclusion map


In mathematics, if is a subset of, then the inclusion map 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; thus:
This and other analogous injective functions 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.

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 is a strong deformation retract of, the inclusion map yields an isomorphism between all homotopy groups.
Inclusion maps in geometry come in different kinds: for example embeddings of submanifolds. Contravariant objects 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.