In mathematics, if is a subset of then the inclusion map is the function that sends each element of to treated as an element of
An inclusion map may also be referred to as an inclusion function, an insertion,^{[1]} or a canonical injection.
A "hooked arrow" (U+21AA ↪ RIGHTWARDS ARROW WITH HOOK)^{[2]} is sometimes used in place of the function arrow above to denote an inclusion map; thus:
(However, some authors use this hooked arrow for any embedding.)
This and other analogous injective functions^{[3]} from substructures are sometimes called natural injections.
Given any morphism between objects and , if there is an inclusion map into the domain , then one can form the restriction of In many instances, one can also construct a canonical inclusion into the codomain known as the range of
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Inclusionexclusion principle made easy

Semester 4 Ring Theory and Linear Algebra 1 Characteristics

Lesson 1 : Mathematical expression
Transcription
Applications of inclusion maps
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 to require that is simply to say that is consistently computed in the substructure 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 is a strong deformation retract of 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 and may be different morphisms, where is a commutative ring and is an ideal of
See also
 Cofibration – continuous mapping between topological spaces
 Identity function – In mathematics, a function that always returns the same value that was used as its argument
References
 ^ MacLane, S.; Birkhoff, G. (1967). Algebra. Providence, RI: AMS Chelsea Publishing. p. 5. ISBN 0821816462.
Note that "insertion" is a function S → U and "inclusion" a relation S ⊂ U; every inclusion relation gives rise to an insertion function.
 ^ "Arrows – Unicode" (PDF). Unicode Consortium. Retrieved 20170207.
 ^ Chevalley, C. (1956). Fundamental Concepts of Algebra. New York, NY: Academic Press. p. 1. ISBN 0121720500.