Coxtype

Coxtype is a theoretical construct in type theory and category theory used to denote the dual notion to algebraic data types. Whereas algebraic data types are defined by constructors that build values, co-types are characterized by destructors that observe or decompose values. In this dual view, a co-type emphasizes the observable structure of data rather than its construction.

A co-type is often associated with coalgebras and final coalgebras, and commonly features corecursive definitions. Instances are described by how they can be observed through a finite set of destructors, and potentially by recursive relationships between observations. This makes co-types well-suited to model infinite or lazily produced data such as streams or reactive processes.

Example: a stream of values of type A can be treated as a co-type with destructors head that yields the next element and tail that yields the remainder stream. In a language supporting codata, such destructors are the primary means of interacting with a stream, rather than constructing a particular stream upfront.

Usage and status: Co-type theory remains a topic of academic research within type theory and category theory. It informs discussions of corecursion, bisimulation, and dualities between data construction and observation. It is not a standard feature in mainstream programming languages, though related ideas appear in languages that support codata, copatterns, or lazy coalgebraic definitions.

See also: algebraic data type, cotype, coalgebra, codata, corecursion, final coalgebra.