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.
One-to-one correspondence is an alternate name for a bijection. For instance, in projective geometry the mappings are correspondences 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, 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.
In category theory, a correspondence from
C
to
D
is a functor
C
o
p
×
D
→
S
e
t
. 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
. 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.