Correspondence (mathematics)

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, i.e. 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.^[4]

• In category theory, a correspondence from $C$ to $D$ is a functor $C^{op}\times D\to\mathbf{Set}$. It is the "opposite" of a profunctor.
• In von Neumann algebra theory, a correspondence is a synonym for a von Neumann algebra bimodule.
• In economics, a correspondence between two sets A and B is a map f:A→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.

See also

• Binary relation
• Bijection

References

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] 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 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.