exponential object

English

The exponential object $Z^{Y}$ (with evaluation morphism ${\text{eval}}:Z^{Y}\times Y\rightarrow Z$) indexes morphisms from Y to Z in a universal way, which means that for a family of morphisms from Y to Z indexed by X (with evaluation morphism $g:X\times Y\rightarrow Z$), there is a unique morphism $\lambda g:X\rightarrow Z^{Y}$, called the transpose of g, such that g factors through ${\text{eval}}$ with cofactor $\lambda g\times {\text{id}}_{Y}$.

Noun

exponential object (plural exponential objects)

1. (category theory) An object which indexes a family of arrows between two given objects in a universal way, meaning that any other indexed family of arrows between the same given pair of objects must factor uniquely through this universally-indexed family of arrows.

   An exponential object generalizes its interpretation in category $\mathbf {Set}$; namely, that of as a function set or internal hom-set.

   The pair $Z^{Y},{\mbox{eval}}:Z^{Y}\times Y\rightarrow Z$ is the terminal object of the comma category $(-\times Y)\downarrow Z$. Therefore the exponential object is a kind of universal morphism.

Hypernyms

• internal Hom

See also

• currying $\sim$ exportation_(logic) (an instance of which is the transpositioning $\lambda$ in the figure)
• modus ponens (homologous to the universal evaluation morphism, ${\text{eval}}$)