exponential object

Noun

 * 1)  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