Editing Currying (section)

=== Set theory ===
In [[set theory]], the notation <math>Y^X</math> is used to denote the [[set (mathematics)|set]] of functions from the set <math>X</math> to the set <math>Y</math>. Currying is the [[natural equivalence|natural bijection]] between the set <math>A^{B\times C}</math> of functions from <math>B\times C</math> to <math>A</math>, and the set <math>(A^C)^B</math> of functions from <math>B</math> to the set of functions from <math>C</math> to <math>A</math>. In symbols:

:<math>A^{B\times C}\cong (A^C)^B</math>

Indeed, it is this natural bijection that justifies the [[exponential notation]] for the set of functions. As is the case in all instances of currying, the formula above describes an [[Adjoint functors|adjoint pair of functors]]: for every fixed set <math>C</math>, the functor <math>B\mapsto B\times C</math> is left adjoint to the functor <math>A \mapsto A^C</math>.

In the [[category of sets]], the [[Mathematical object|object]] <math>Y^X</math> is called the [[exponential object]].