# Correspondence (mathematics)

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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*, i.e. any subset of the Cartesian product*X*×*Y*.^{[1]}

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- 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,
^{[2]}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.^{[3]}

- In category theory, a
**correspondence**from to is a functor . It is the "opposite" of a profunctor.

is an alternate name for a bijection.**One-to-one correspondence**

- 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*. 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.