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In mathematics and mathematical economics, correspondence is a term with several related but distinct meanings.

In general mathematics, a correspondence is an ordered triple (X,Y,R), where R is a relation from X to Y, that is, any subset of the Cartesian product X×Y.^[1]
One-to-one correspondence is an alternate name for a bijection. For instance, in projective geometry the mappings are correspondences^[2] between projective ranges.

In algebraic geometry, a correspondence between algebraic varieties V and W is in the same fashion a subset R of V×W, which is in addition required to be closed in the Zariski topology. It therefore means any relation that is defined by algebraic equations. There are some important examples, even when V and W are algebraic curves: for example the Hecke operators of modular form theory may be considered as correspondences of modular curves.

However, the definition of a correspondence in algebraic geometry is not completely standard. For instance, Fulton, in his book on Intersection theory,^[3] uses the definition above. In literature, however, a correspondence from a variety X to a variety Y is often taken to be a subset Z of X×Y such that Z is finite and surjective over each component of X. Note the asymmetry in this latter definition; which talks about a correspondence from X to Y rather than a correspondence between X and Y. The typical example of the latter kind of correspondence is the graph of a function f:X→Y. Correspondences also play an important role in the construction of motives (cf. presheaf with transfers).^[4]

In category theory, a correspondence from $C$ to $D$ is a functor $C^{\text{op}}\times D\to \mathbf {Set}$ . It is the "opposite" of a profunctor.^{[citation needed]}
In von Neumann algebra theory, a correspondence is a synonym for a von Neumann algebra bimodule.^{[citation needed]}
In economic, a correspondence between two sets $A$ and $B$ is a map $f:A\to P(B)$ from the elements of the set $A$ to the power set of $B$ .^[5] This is similar to a correspondence as defined in general mathematics (i.e., a relation,) except that the range is over sets instead of elements. However, there is usually the additional property that for all a in A, f(a) is not empty. In other words, each element in A maps to a non-empty subset of B; or in terms of a relation R as subset of A×B, R projects to A surjectively. A correspondence with this additional property is thought of as the generalization of a function, rather than as a special case of a relation, and is referred to in other contexts as a multivalued function.

An example of a correspondence in this sense is the best response correspondence in game theory, which gives the optimal action for a player as a function of the strategies of all other players. If there is always a unique best action given what the other players are doing, then this is a function. If for some opponent's strategy, there is a set of best responses that are equally good, then this is a correspondence.

References

^ Encyclopedic dictionary of Mathematics. MIT. 2000. pp. 1330–1331. ISBN 0-262-59020-4.
^ H. S. M. Coxeter (1959) The Real Projective Plane, page 18
^ Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98549-7, MR 1644323
^ Mazza, Carlo; Voevodsky, Vladimir; Weibel, Charles (2006), Lecture notes on motivic cohomology, Clay Mathematics Monographs, vol. 2, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3847-1, MR 2242284
^ Mas-Colell, Andreu; Whinston, Michael D.; Green, Jerry R. (1995). Microeconomic Analysis. New York: Oxford University Press. pp. 949–951. ISBN 0-19-507340-1.

This article includes a list of related items that share the same name (or similar names).
If an internal link incorrectly led you here, you may wish to change the link to point directly to the intended article.

[1] Encyclopedic dictionary of Mathematics. MIT. 2000. pp. 1330–1331. ISBN 0-262-59020-4.

[2] H. S. M. Coxeter (1959) The Real Projective Plane, page 18

[3] Fulton, William (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98549-7, MR 1644323

[4] Mazza, Carlo; Voevodsky, Vladimir; Weibel, Charles (2006), Lecture notes on motivic cohomology, Clay Mathematics Monographs, vol. 2, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-3847-1, MR 2242284

[5] Mas-Colell, Andreu; Whinston, Michael D.; Green, Jerry R. (1995). Microeconomic Analysis. New York: Oxford University Press. pp. 949–951. ISBN 0-19-507340-1.

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+In mathematics and mathematical economics, '''correspondence''' is a term with several related but distinct meanings.
-#redirect [[binary relation#other uses of correspondence]]
+* In general [[mathematics]], a '''correspondence''' is an ordered triple (''X'',''Y'',''R''), where ''R'' is a [[binary relation|relation]] from ''X'' to ''Y'', that is, any [[subset]] of the [[Cartesian product]] ''X''×''Y''.<ref>{{cite book |title=Encyclopedic dictionary of Mathematics |publisher=MIT |pages=1330–1331 |year=2000 |isbn=0-262-59020-4 |url=https://books.google.com/books?id=azS2ktxrz3EC&pg=PA1331#v=onepage&f=false }}</ref>
+* '''''One-to-one correspondence''''' is an alternate name for a [[bijection]]. For instance, in [[projective geometry]] the [[Map (mathematics)|mappings]] are ''correspondences''<ref>[[H. S. M. Coxeter]] (1959) ''The Real Projective Plane'', page 18</ref> between [[projective range]]s.
+{{anchor|algebraic geometry}}
+* In [[algebraic geometry]], a '''correspondence''' between [[algebraic variety|algebraic varieties]] ''V'' and ''W'' is in the same fashion a subset ''R'' of ''V''×''W'', which is in addition required to be closed in the [[Zariski topology]]. It therefore means any relation that is defined by algebraic equations. There are some important examples, even when ''V'' and ''W'' are [[algebraic curve]]s: for example the [[Hecke operator]]s of [[modular form]] theory may be considered as correspondences of [[modular curve]]s.
+:However, the definition of a correspondence in algebraic geometry is not completely standard. For instance, Fulton, in his book on [[Intersection theory]],<ref>{{Citation | last1=Fulton | first1=William | author1-link=William Fulton (mathematician) | title=Intersection theory | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] | isbn=978-0-387-98549-7 | mr=1644323  | year=1998 | volume=2}}</ref> uses the definition above. In literature, however, a correspondence from a variety ''X'' to a variety ''Y'' is often taken to be a subset ''Z'' of ''X''×''Y'' such that ''Z'' is finite and surjective over each component of ''X''. Note the asymmetry in this latter definition; which talks about a correspondence from ''X'' to ''Y'' rather than a correspondence between ''X'' and ''Y''. The typical example of the latter kind of correspondence is the graph of a function ''f'':''X''→''Y''. Correspondences also play an important role in the construction of [[motive (algebraic geometry)|motives]] (cf. [[presheaf with transfers]]).<ref>{{Citation | last1=Mazza | first1=Carlo | last2=Voevodsky | first2=Vladimir | author2-link=Vladimir Voevodsky | last3=Weibel | first3=Charles | title=Lecture notes on motivic cohomology | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=[[Clay Mathematics Monographs]] | isbn=978-0-8218-3847-1 | mr=2242284  | year=2006 | volume=2}}</ref>
+* In [[category theory]], a '''correspondence''' from <math> C </math> to <math> D </math> is a functor <math>C^\text{op}\times D\to\mathbf{Set}</math>. It is the "opposite" of a [[profunctor]].{{fact|date=January 2019}}
+* In von Neumann algebra theory, a '''correspondence''' is a synonym for a [[von Neumann algebra]] bimodule.{{fact|date=January 2019}}
+* In [[economics|economic]], a '''correspondence''' between two sets <math>A</math> and <math>B</math> is a [[map (mathematics)|map]] <math>f: A \to P(B)</math> from the elements of the set <math>A</math> to the [[power set]] of <math>B</math>.<ref>{{cite book |last=Mas-Colell |first=Andreu |authorlink=Andreu Mas-Colell |last2=Whinston |first2=Michael D. |last3=Green |first3=Jerry R. |title=Microeconomic Analysis |location=New York |publisher=Oxford University Press |year=1995 |isbn=0-19-507340-1 |pages=949–951 |url=https://books.google.com/books?id=KGtegVXqD8wC&pg=PA949 }}</ref>  This is similar to a correspondence as defined in general mathematics (i.e., a [[Relation (mathematics)|relation]],) except that the range is over sets instead of elements.  However, there is usually the additional property that for all ''a'' in ''A'', ''f''(''a'') is not empty.  In other words, each element in ''A'' maps to a non-empty subset of ''B''; or in terms of a relation ''R'' as subset of ''A''×''B'', ''R'' projects to ''A'' [[surjective]]ly.  A correspondence with this additional property is thought of as the generalization of a [[function (mathematics)|function]], rather than as a special case of a relation, and is referred to in other contexts as a [[multivalued function]].
+:An example of a correspondence in this sense is the [[best response]] correspondence in [[game theory]], which gives the optimal action for a player as a function of the strategies of all other players.  If there is always a unique best action given what the other players are doing, then this is a function.  If for some opponent's strategy, there is a set of best responses that are equally good, then this is a correspondence.
+==See also==
+* [[Heterogeneous relation]]
+==References==
+{{Reflist}}
+[[Category:Mathematical terminology]]
+{{Set index article}}