currying

Currying is the technique of emulating multiple-parameteredfunctions with higher-order functions. The notion is that a functionof $n$ arguments can be thought of as a function of 1 argument thatmaps to a function of $n-1$ arguments. A curried function is afunction represented by currying, e.g.

$$f:A\to (B\to C)$$

For conciseness, the mapping operator $\to$ is usuallyconsidered right-associative, so one could drop the parentheses in theexpression above and write $f:A\to B\to C$ instead.

In contrast, an uncurried function is usually specified as amapping from a Cartesian product, such as

$$f:(A\times B)\to C.$$

The term currying is derived from the name of Haskell Curry, a20th-century logician. However, Curry was not the first person todiscover this notion, as it was first introduced by Gottlob Frege in1893 and expanded by Moses Schönfinkel in the 1920s. Hence thenotion is sometimes referred to as schönfinkeling.

From the perspective of category theory, currying can be thought of asexploiting the fact that $-\times B$ and $\mathrm{Hom}(B,-)$ are adjointfunctors on $\mathrm{𝐒𝐞𝐭}$. That is, for each set $B$, there is a naturalequivalence

$$\nu :{\mathrm{Hom}}_{\mathrm{𝐒𝐞𝐭}}(-\times B,-)\stackrel{\cdot }{⟶}{\mathrm{Hom}}_{\mathrm{𝐒𝐞𝐭}}(-,\mathrm{Hom}(B,-))$$

defined by sending a map $f:(A\times B)\to C$ to the map${\nu }_f:A\to \mathrm{Hom}(B,C)$. For each $a\in A$, ${\nu }_f(a):B\to C$ is the map defined by ${\nu }_f(a)(b)=f(a,b)$.