Editing Inclusion map (section)

==Applications of inclusion maps==
Inclusion maps tend to be [[homomorphism]]s of [[algebraic structure]]s; thus, such inclusion maps are [[embedding]]s.  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 <math>\star,</math> to require that
<math display=block>\iota(x\star y) = \iota(x) \star \iota(y)</math>
is simply to say that <math>\star</math> 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 (mathematics)|closure]] means such constants must already be given in the substructure.

Inclusion maps are seen in [[algebraic topology]] where if <math>A</math> is a [[strong deformation retract]] of <math>X,</math> the inclusion map yields an [[Group isomorphism|isomorphism]] between all [[homotopy groups]] (that is, it is a [[Homotopy|homotopy equivalence]]).

Inclusion maps in [[geometry]] come in different kinds: for example [[embedding]]s of [[submanifold]]s. [[Covariance and contravariance of functors|Contravariant]] objects (which is to say, objects that have [[pullback]]s; these are called [[covariance and contravariance of vectors|covariant]] in an older and unrelated terminology) such as [[differential form]]s ''restrict'' to submanifolds, giving a mapping in the ''other direction''. Another example, more sophisticated, is that of [[affine scheme]]s, for which the inclusions
<math display=block>\operatorname{Spec}\left(R/I\right) \to \operatorname{Spec}(R)</math>
and
<math display=block>\operatorname{Spec}\left(R/I^2\right) \to \operatorname{Spec}(R)</math>
may be different [[morphism]]s, where <math>R</math> is a [[commutative ring]] and <math>I</math> is an [[Ideal (ring theory)|ideal]] of <math>R.</math>