Inclusion map

File:Venn A subset B.svg

$A$ 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 is the function $\iota$ that sends each element $x$ of $A$ to $x,$ treated as an element of $B:$ $$\iota :A\to B,\iota (x)=x.$$ 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 $f.$

Applications 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 $$\mathrm{Spec}(R/I)\to \mathrm{Spec}(R)$$ and $$\mathrm{Spec}(R/I^2)\to \mathrm{Spec}(R)$$ may be different morphisms, where $R$ is a commutative ring and $I$ is an ideal of $R.$

See also

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