# exponential object

## English

The exponential object ZY — which may be pronounced "Z by Y" — indexes a family of arrows from Y to Z in a universal way, which means that for any X-indexed family of arrows g from Y to Z, g must factor through eval by means of an unique other arrow which is the product of the transpose of g and the identity of 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 ${\displaystyle \mathbf {Set} }$; namely, that of as a function set or internal hom-set.
The pair ${\displaystyle Z^{Y},{\mbox{eval}}:Z^{Y}\times Y\rightarrow Z}$ is the terminal object of the comma category ${\displaystyle (-\times Y)\downarrow Z}$. Therefore the exponential object is a kind of universal morphism.