exponential objectとは 意味・読み方・使い方

名詞

exponential object (複数形 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.

上位語

• internal Hom

参考

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