In mathematics is a subset then the inclusion map is the function \iota

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

A "hooked arrow" ()

(However, some authors use this hooked arrow for any embedding.)

This and other analogous injective functions from substructures are sometimes called natural injections.

Given any morphism between objects into the domain, then one can form the restriction In many instances, one can also construct a canonical inclusion into the codomain known as the range

Applications of inclusion maps

Inclusion maps tend to be homomorphisms of algebraic structures; thus, such inclusion maps are embeddings 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 is a strong deformation retract 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 may be different morphisms is a commutative ring is an ideal

See also

References