Exponential

Exponential function(Q168698) is a construction or function that can be represented as the following category:

In the category of sets, the morphisms between sets X and Y are the functions from X to Y. It results that the set of the functions from X to Y that is denoted $Y^{X}$ in the preceding section can also be denoted $\hom(X,Y).$ The isomorphism $(S^{T})^{U}\cong S^{T\times U}$ can be rewritten

$\hom(U,S^{T})\cong \hom(T\times U,S).$

This means the functor "exponentiation to the power T " is a right adjoint to the functor "direct product with T ".

This generalizes to the definition of exponentiation in a category in which finite direct products exist: in such a category, the functor $X\to X^{T}$ is, if it exists, a right adjoint to the functor $Y\to T\times Y.$ A category is called a Cartesian closed category, if direct products exist, and the functor $Y\to X\times Y$ has a right adjoint for every T.