Kindcategory

Kindcategory is a term used in programming language theory to describe a categorical framework for organizing kinds, the classifications of type constructors in a language with higher-kinded types. In this view, kinds are treated as objects of a category, and morphisms represent admissible transformations between kinds, such as mappings induced by type constructors or polymorphic type-level functions. The framework allows formal reasoning about how kinds compose and interact, and provides a language for expressing how more complex kinds are built from simpler ones.

Common examples of objects include the star kind (*) for ordinary types and constructions like * -> * for unary type constructors. In a kindcategory, a morphism might model a way to map an n-ary kind to an m-ary kind, or capture a functorial relationship between type constructors. Composition of morphisms corresponds to composing these kind-level transformations, while identity morphisms preserve a given kind.

Applications of the concept include reasoning about higher-kinded types, parametric polymorphism, and the design and verification of type systems and compilers that support higher-kinded or dependent types. Some variants explore enriched or dependent versions, where morphisms depend on terms or contexts.

Kindcategory is not a universally standardized term; definitions vary by author and formalism. It serves as a conceptual tool to compare and relate different approaches to kinds and higher-kinded type systems.

See also: category theory; higher-kinded types; kind (type theory); polymorphism.