Inclusion map

In mathematics, an inclusion map is a function i of the particular form

i:Y → X, i(x) = x

where X is a set and Y is a subset of X. The inclusion relation on the power set of X is the partial order on subsets of X given by set containment. These fundamental concepts may be redescribed in many ways. For example one can note that the inclusion partial order is (up to isomorphism) the direct product of |X| copies of the partial order on {0,1} for which 0 < 1.

Inclusion maps tend to be homomorphisms of algebraic structures; more precisely, given a sub-structure closed under some operations, the inclusion map will be a homomorphism for tautological reasons, given the very definition by restriction of what one checks. For example, for a binary operation @, to require that

i(x@y) = i(x)@i(y)

is simply to say that @ 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 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 a ring ideal.