Inclusion map

 ( Basic Concepts In Set Theory ) 

In mathematics, if $A$ is a subset of $B,$ then the inclusion map is the function $\backslash iota$ that sends each element $x$ of $A$ to $x,$ treated as an element of $B:$ $$\backslash iota\; :\; A\backslash rightarrow\; B,\; \backslash qquad\; \backslash iota(x)=x.$$

An inclusion map may also be referred to as an inclusion function, an insertion,

or a canonical injection.

A "hooked arrow" () is sometimes used in place of the function arrow above to denote an inclusion map; thus: $$\backslash iota:\; A\backslash hookrightarrow\; B.$$

(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 $\backslash iota\; :\; A\; \backslash to\; X$ into the domain $X$, then one can form the restriction $f\backslash circ\; \backslash iota$ of $f.$ In many instances, one can also construct a canonical inclusion into the codomain $R\; \backslash to\; Y$ known as the range of $f.$

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Applications of inclusion maps Inclusion maps tend to be of algebraic structures; thus, such inclusion maps are . 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 $\backslash star,$ to require that $$\backslash iota(x\backslash star\; y)\; =\; \backslash iota(x)\; \backslash star\; \backslash iota(y)$$ is simply to say that $\backslash star$ 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).

Inclusion maps in geometry come in different kinds: for example of . Contravariant objects (which is to say, objects that have ; 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 , for which the inclusions $$\backslash operatorname\{Spec\}\backslash left(R/I\backslash right)\; \backslash to\; \backslash operatorname\{Spec\}(R)$$ and $$\backslash operatorname\{Spec\}\backslash left(R/I^2\backslash right)\; \backslash to\; \backslash operatorname\{Spec\}(R)$$ may be different , where $R$ is a commutative ring and $I$ is an ideal of $R.$

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See also

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