Inclusion map: Difference between revisions
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{{Short description|Set-theoretic function}} |
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[[File:Venn A subset B.svg|150px|thumb|right| |
[[File:Venn A subset B.svg|150px|thumb|right|<math>A</math> is a [[subset]] of <math>B,</math> and <math>B</math> is a [[Subset|superset]] of <math>A.</math>]] |
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In [[mathematics]], if <math>A</math> is a [[subset]] of <math>B,</math> then the '''inclusion map''' is the [[function (mathematics)|function]] [[ι|<math>\iota</math>]] that sends each element <math>x</math> of <math>A</math> to <math>x,</math> treated as an element of <math>B:</math> |
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⚫ | An inclusion map may also be referred to as an '''inclusion function''', an '''insertion''',<ref>{{cite book| first1 = S. | last1 = MacLane | first2 = G. | last2 = Birkhoff | title = Algebra | publisher = AMS Chelsea Publishing |location=Providence, RI | year = 1967| isbn = 0-8218-1646-2 | page = 5 | quote = Note that “insertion” is a function {{math|''S'' → ''U''}} and "inclusion" a relation {{math|''S'' ⊂ ''U''}}; every inclusion relation gives rise to an insertion function.}}</ref> or a '''canonical injection'''. |
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A "hooked arrow" < |
A "hooked arrow" ({{unichar|21AA|RIGHTWARDS ARROW WITH HOOK|ulink=Unicode}})<ref name="Unicode Arrows">{{cite web| title = Arrows – Unicode| url = https://www.unicode.org/charts/PDF/U2190.pdf| access-date = 2017-02-07|publisher=[[Unicode Consortium]]}}</ref> is sometimes used in place of the function arrow above to denote an inclusion map; thus: |
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<math display=block>\iota: A\hookrightarrow B.</math> |
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(However, some authors use this hooked arrow for any [[embedding]].) |
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This and other analogous [[injective]] functions<ref>{{citation| first = C. | last = Chevalley | title = Fundamental Concepts of Algebra | year = 1956}}, page 1 |
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</ref> from [[substructure]]s are sometimes called '''''natural injections'''''. |
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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]] |
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⚫ | Given any [[morphism]] <math>f</math> between [[object (category theory)|objects]] <math>X</math> and <math>Y</math>, if there is an inclusion map <math>\iota : A \to X</math> into the [[Domain of a function|domain]] <math>X</math>, then one can form the [[Restriction (mathematics)|restriction]] <math>f\circ \iota</math> of <math>f.</math> In many instances, one can also construct a canonical inclusion into the [[codomain]] <math>R \to Y</math> known as the [[range of a function|range]] of <math>f.</math> |
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⚫ | Inclusion maps tend to be [[homomorphism]]s of [[algebraic structure]]s; thus, such inclusion maps are [[embedding]]s. More precisely, given a |
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⚫ | Inclusion maps tend to be [[homomorphism]]s of [[algebraic structure]]s; thus, such inclusion maps are [[embedding]]s. 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 <math>\star,</math> to require that |
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⚫ | is simply to say that <math>\star</math> 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 (mathematics)|closure]] means such constants must already be given in the substructure. |
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⚫ | is simply to say that <math>\star</math> 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 [[ |
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⚫ | Inclusion maps in [[geometry]] come in different kinds: for example [[embedding]]s of [[submanifold]]s. [[Covariance and contravariance of functors|Contravariant]] objects |
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⚫ | Inclusion maps in [[geometry]] come in different kinds: for example [[embedding]]s of [[submanifold]]s. [[Covariance and contravariance of functors|Contravariant]] objects (which is to say, objects that have [[pullback]]s; these are called [[covariance and contravariance of vectors|covariant]] in an older and unrelated terminology) such as [[differential form]]s ''restrict'' to submanifolds, giving a mapping in the ''other direction''. Another example, more sophisticated, is that of [[affine scheme]]s, for which the inclusions |
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and |
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==See also== |
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* {{annotated link|Cofibration}} |
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== |
==References== |
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*[[Cofibration]] |
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==Notes== |
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{{reflist}} |
{{reflist}} |
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== References == |
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* {{citation| first = C. | last = Chevalley | title = Fundamental Concepts of Algebra | publisher = Academic Press, New York | year = 1956| isbn = 0-12-172050-0}}. |
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* {{citation| first1 = S. | last1 = Mac Lane | first2 = G. | last2 = Birkhoff | title = Algebra | publisher = AMS Chelsea Publishing, Providence, Rhode Island | year = 1967| isbn = 0-8218-1646-2}}. |
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[[Category:Functions and mappings]] |
[[Category:Functions and mappings]] |
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Latest revision as of 00:37, 27 September 2024
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
Applications of inclusion maps
[edit]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 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
[edit]- 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
[edit]- ^ 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.