exponential object

English

The exponential object $Z^Y$ (with evaluation morphism $\text{eval}:Z^Y\times Y\to Z$) 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 $g:X\times Y\to Z$), there is a unique morphism $\lambda g:X\to Z^Y$, 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 $𝐒𝐞𝐭$; namely, that of as a function set or internal hom-set.

   The pair $Z^Y,\text{eval}:Z^Y\times Y\to Z$ 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, $\text{eval}$)