inclusion map について

is sometimes used in place of the function arrow above to denote an inclusion map.
This and other analogous injective functions 〔, page 1
〕 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 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
:''Spec(R/I)'' → ''Spec(R)''
and
:''Spec(R/I²)'' → ''Spec(R)''
may be different morphisms, where ''R'' is a commutative ring and ''I'' an ideal.

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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