Typed lambda calculi

In this context, types are usually objects of a syntactic nature that are assigned to lambda terms; the exact nature of a type depends on the calculus considered.

From a certain point of view, typed lambda calculi can be seen as refinements of the untyped lambda calculus, but from another point of view, they can also be considered the more fundamental theory where the untyped lambda calculus is just a special case with a single (all-encompassing, universal) type.

Typed lambda calculi are foundational programming languages and are the base of typed functional programming languages such as ML and Haskell and, more indirectly, typed imperative programming languages.

Typed lambda calculi play an important role in the design of type systems for programming languages; here typability usually captures desirable properties of the program (e.g. the program will not cause a memory access violation).

Typed lambda calculi are closely related to mathematical logic and proof theory via the Curry-Howard isomorphism and they can be considered as the internal language of classes of categories, e.g. the simply typed lambda calculus is the language of cartesian closed categories.

The simply typed lambda calculus is the language of Cartesian Closed Categories.