Type constructor

In the area of mathematical logic and computer science known as type theory, a type constructor is a feature of a typed formal language that builds new types from old ones. Basic types are considered to be built using nullary type constructors. Some type constructors take another type as an argument, e.g., the constructors for product types, function types, power types and list types. New types can be defined by recursively composing type constructors.

For example, simply typed lambda calculus can be seen as a language with a single type constructor—the function type constructor. Product types can generally be considered "built-in" in typed lambda calculi via currying.

Abstractly, a type constructor is an n-ary type operator taking as argument zero or more types, and returning another type. Making use of currying, n-ary type operators can be (re)written as a sequence of applications of unary type operators. Therefore, we can view the type operators as a simply typed lambda calculus, which has only one basic type, usually denoted , and pronounced "type", which is the type of all types in the underlying language, which are now called proper types in order to distinguish them from the types of the type operators in their own calculus, which are called kinds.

Instituting a simply typed lambda calculus over the type operators results in more than just a formalization of type constructors though. Higher-order type operators become possible. Type operators correspond to the 2nd axis in the lambda cube, leading to the simply typed lambda-calculus with type operators, λω; while this is not so well known, combining type operators with polymorphic lambda calculus (system F) yields system F-omega.

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