kindtype

Kindtype is a term used in discussions of type systems to denote a classification of type constructors at a level above ordinary types. It is not a widely standardized feature in mainstream languages, but it appears in theoretical treatments of higher-kinded types and universe hierarchies. In such contexts, a kind classifies either a type or a type constructor; a kindtype would then operate at the level of classifying kinds themselves, effectively serving as a meta-classification for type-level abstractions.

Definition and usage: A kindtype can be thought of as a type whose instances are kinds or type constructors, rather than ordinary data. It provides a way to talk about and manipulate the shapes of types in a language that permits higher-kinds and universe polymorphism. In practice, discussions of kindtypes are often part of theoretical explorations rather than concrete language features.

Relation to existing concepts: In languages with higher-kinds, concepts such as Type, kinds, and universes (for example Type0, Type1) arise. A kindtype might be described as the type of these kind-like objects, or as a universe for kinds. This framing emphasizes the meta-level role of kinds in organizing the type system rather than introducing a specific syntax.

Examples: In a hypothetical language, a kindtype K could have values representing the kinds Star, Arrow, and their compositions, enabling type constructors to be abstracted over different kinds. Such a construct would support higher-order abstractions and more expressive type-level programming.

See also: Kind, Type, Higher-kinded types, Type theory, Universe (type theory), Sort (type theory).