Inclusion map: Difference between revisions
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(On the other hand, this notation is sometimes reserved for [[embedding]]s.) |
(On the other hand, this notation is sometimes reserved for [[embedding]]s.) |
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This and other analogous [[injective]] functions<ref>{{cite book| first = C. | last = Chevalley | title = Fundamental Concepts of Algebra | publisher = Academic Press|location= New York, NY | year = 1956| isbn = 0-12-172050-0 |page= 1}}</ref> from [[substructure (mathematics)|substructures]] are sometimes called '''natural injections'''. |
This and other analogous [[injective]] functions<ref>{{cite book| first = C. | last = Chevalley | title = Fundamental Concepts of Algebra | url = https://archive.org/details/fundamentalconce00chev_0 | url-access = registration | publisher = Academic Press|location= New York, NY | year = 1956| isbn = 0-12-172050-0 |page= [https://archive.org/details/fundamentalconce00chev_0/page/1 1]}}</ref> from [[substructure (mathematics)|substructures]] are sometimes called '''natural injections'''. |
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Given any [[morphism]] {{mvar|f}} between [[object (category theory)|objects]] {{mvar|X}} and {{mvar|Y}}, if there is an inclusion map into the [[domain (mathematics)|domain]] {{math|''ι'' : ''A'' → ''X''}}, then one can form the [[Function (mathematics)#Restrictions and extensions|restriction]] {{mvar|f ι}} of {{mvar|f}}. In many instances, one can also construct a canonical inclusion into the [[codomain]] {{math|''R'' → ''Y''}} known as the [[range (mathematics)|range]] of {{mvar|f}}. |
Given any [[morphism]] {{mvar|f}} between [[object (category theory)|objects]] {{mvar|X}} and {{mvar|Y}}, if there is an inclusion map into the [[domain (mathematics)|domain]] {{math|''ι'' : ''A'' → ''X''}}, then one can form the [[Function (mathematics)#Restrictions and extensions|restriction]] {{mvar|f ι}} of {{mvar|f}}. In many instances, one can also construct a canonical inclusion into the [[codomain]] {{math|''R'' → ''Y''}} known as the [[range (mathematics)|range]] of {{mvar|f}}. |
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Revision as of 04:30, 9 November 2019
In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion,[1] or canonical injection) is the function ι that sends each element x of A to x, treated as an element of B:
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:
(On the other hand, this notation is sometimes reserved for embeddings.)
This and other analogous injective functions[3] 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 ι : A → X, then one can form the restriction f ι 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 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 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
and
may be different morphisms, where R is a commutative ring and I is an ideal of R.
See also
References
- ^ MacLane, S.; Birkhoff, G. (1967). Algebra. Providence, RI: AMS Chelsea Publishing. p. 5. ISBN 0-8218-1646-2.
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 2017-02-07.
- ^ Chevalley, C. (1956). Fundamental Concepts of Algebra. New York, NY: Academic Press. p. 1. ISBN 0-12-172050-0.
External links
Media related to Inclusion map at Wikimedia Commons