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}

.

exponential object (plural exponential objects)

(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.