What is "exponential object"

n. (context category theory English) A categorical object which generalizes its interpretation in category '''Set'''; namely, as a function set's article on (Revision 237). An exponential may be introduced through the "http://en.wikipedi

1. org/wiki/Currying" inference ruleJeltsch, Wolfgang (2012). , slide 20. $\{f\; :\; C\; times\; A\; rightarrow\; B\; over\; lambda\_f\; :\; C\; rightarrow\; B^A\}$ and eliminated through the "function http://en.wikipedia.org/wiki/Apply" ("eval") rule $epsilon\_(\}\_\{A,\; B)\; :\; B^A\; times\; A\; rightarrow\; B$.

In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed categories.