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. ==See also==