typem

Typem is a theoretical construct used in discussions of type theory and metaprogramming to denote the type of types. In languages with facilitating reflection or type-level computation, a typem value stands for a type itself rather than for an instance of that type. This allows programs to manipulate types as first-class entities, enabling generic abstractions, type-level transformations, and dependent programming patterns.

Semantics: A typem value often carries metadata about a type, such as its arity, constraints, or constructors. In a language with a sound kind system, typem may inhabit a universe of kinds, and operations on typems must respect termination and consistency constraints. Depending on the design, typem may be treated as a primitive, a built-in kind, or a parametric abstraction over other types.

Usage: Typem is used to express generic operations that produce or modify types, to implement type-safe factories, and to encode metafunctions. For example, a function derive could accept a typem and yield a new type by applying a transformation to the type’s structure. Typem can also enable type-level maps, higher-kinded polymorphism, or dependent types, depending on the language.

Limitations: The inclusion of typem raises questions about type safety, impredicativity, and decidability of type checking. Some systems restrict typem to prevent paradoxes, while others rely on stratified universes or explicit annotations to preserve soundness.

See also: Type theory, Type systems, Metaprogramming, Kind systems.