Inclusion map

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is a subset of and is a superset of

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

An inclusion map may also 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.

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[edit]

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 (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

may be different morphisms, where is a commutative ring and is an ideal of

See also[edit]

  • Cofibration – continuous mapping between topological spaces
  • Identity function – In mathematics, a function that always returns the same value that was used as its argument


  1. ^ MacLane, S.; Birkhoff, G. (1967). Algebra. Providence, RI: AMS Chelsea Publishing. p. 5. ISBN 0-8218-1646-2. Note that "insertion" is a function SU and "inclusion" a relation SU; every inclusion relation gives rise to an insertion function.
  2. ^ "Arrows – Unicode" (PDF). Unicode Consortium. Retrieved 2017-02-07.
  3. ^ Chevalley, C. (1956). Fundamental Concepts of Algebra. New York, NY: Academic Press. p. 1. ISBN 0-12-172050-0.