From Basic Category Theory for Computer Scientists

Let $C$ be a category with all binary products and let $A$ and $B$ be objects of $C$. An object $B^A$ is an exponential object if there is an arrow $eval_{AB} : (B^A \times A) \to B$ such that for any object $C$ and arrow $g: (C \times A)\to B$ there is a unique arrow $curry(g): C \to B^A$ such that $eval_{AB} \circ (curry(g) \times id_A) =g$.

enter image description here

Can exponential object be defined in terms of universal construction?

A universal construction describes a class of objects and accompanying arrows that share a common property and picks out the objects that are terminal when this class is considered as a category.

  • 3
    $\begingroup$ What do you mean ? This is a definition by a universal property $\endgroup$ Jul 19, 2019 at 21:21

1 Answer 1


Yes: take the category in which an object is a pair $(X,f)$ where $X$ is an object of $C$ and $f:X\times A\to B$, and a morphism between two such objects $(X,f)$ and $(Y,g)$ is an arrow $h:X\to Y$ such that $g\circ (h\times id_A)=f$. Your given definition then says exactly that an exponential object $(B^A,eval_{AB})$ is a terminal object in this category.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.