Inclusion map

A is a subset of B, and B is a superset of A.

In mathematics, if $A$ is a subset of $B$, then the inclusion map (also inclusion function, insertion,^[1] or canonical injection) is the function $\iota$ that sends each element, $x$, of $A$ to $x$, treated as an element of $B$:

$\iota :A\rightarrow B,\qquad \iota (x)=x.$

A "hooked arrow" $\hookrightarrow$ is sometimes used in place of the function arrow above to denote an inclusion map.

This and other analogous injective functions^[2] from substructures are sometimes called natural injections.

Given any morphism f between objects X and Y, if there is an inclusion map into the domain $\iota :A\rightarrow X$, then one can form the restriction fi of f. In many instances, one can also construct a canonical inclusion into the codomain R→Y known as the range of f.

Applications of inclusion maps

Inclusion maps tend to be homomorphisms of algebraic structures; thus, such inclusion maps are embeddings. More precisely, given a sub-structure closed under some operations, the inclusion map will be an embedding for tautological reasons. For example, for a 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 (i.e. is a homotopy equivalence).

Inclusion maps in geometry come in different kinds: for example embeddings of submanifolds. Contravariant objects^[3] 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

$\operatorname {Spec}(R/I)\to \operatorname {Spec}(R)$

and

${\displaystyle \operatorname {Spec} (R/I^{2})\to \operatorname {Spec} (R)}$

may be different morphisms, where R is a commutative ring and I an ideal.

See also

• Cofibration
• Identity function

Notes

1. Mac Lane, S.; Birkhoff, G. (1967), Algebra, page 5, says "Note that “insertion” is a function ${\displaystyle S\to U}$ and “inclusion” a relation ${\displaystyle S\subset U}$; every inclusion relation gives rise to an insertion function."
2. Chevalley, C. (1956), Fundamental Concepts of Algebra, page 1
3. I.e., objects that have pullbacks; these are called covariant in an older and unrelated terminology

References

• Chevalley, C. (1956), Fundamental Concepts of Algebra, Academic Press, New York, ISBN 0-12-172050-0.
• Mac Lane, S.; Birkhoff, G. (1967), Algebra, AMS Chelsea Publishing, Providence, Rhode Island, ISBN 0-8218-1646-2.