In mathematics, if is a subset of then the inclusion map (also inclusion function, insertion, 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) 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 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
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
Note that "insertion" is a function S → U and "inclusion" a relation S ⊂ U; every inclusion relation gives rise to an insertion function.