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Inclusion map

Set-theoretic function

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated April 7, 2026
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In mathematics, if A is a subset of B, then the inclusion map is the function \iota that sends each element x of A to x, treated as an element of B: \iota :A\rightarrow B,\qquad \iota (x)=x.

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

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: \iota :A\hookrightarrow B.

(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 f between objects X and Y, if there is an inclusion map \iota :A\to X into the domain X, then one can form the restriction f\circ \iota of f. In many instances, one can also construct a canonical inclusion into the codomain R\to Y known as the range of f.

01Applications 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 \star , to require that \iota (x\star y)=\iota (x)\star \iota (y) is simply to say that \star 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 A is a strong deformation retract of X, 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 \operatorname {Spec} \left(R/I\right)\to \operatorname {Spec} (R) and \operatorname {Spec} \left(R/I^{2}\right)\to \operatorname {Spec} (R) may be different morphisms, where R is a commutative ring and I is an ideal of R.

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Sources and credits

This article is adapted from the Wikipedia article “Inclusion map”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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