Editing Currying (section)

=== Type theory ===
In [[type theory]], the general idea of a [[type system]] in computer science is formalized into a specific algebra of types. For example, when writing <math>f \colon X \to Y </math>, the intent is that <math>X</math> and <math>Y</math> are [[type system|types]], while the arrow <math>\to</math> is a [[type constructor]], specifically, the [[function type]] or arrow type. Similarly, the Cartesian product <math>X \times Y</math> of types is constructed by the [[product type]] constructor <math>\times</math>.

The type-theoretical approach is expressed in programming languages such as [[ML (programming language)|ML]] and the languages derived from and inspired by it: [[Caml]], [[Haskell]], and [[F Sharp (programming language)|F#]].

The type-theoretical approach provides a natural complement to the language of [[category theory]], as discussed below. This is because categories, and specifically, [[monoidal categories]], have an [[internal language]], with [[simply typed lambda calculus]] being the most prominent example of such a language. It is important in this context, because it can be built from a single type constructor, the arrow type. Currying then endows the language with a natural product type. The correspondence between objects in categories and types then allows programming languages to be re-interpreted as logics (via [[Curry–Howard correspondence]]), and as other types of mathematical systems, as explored further, below.