profunctor

A profunctor is a mathematical concept that combines aspects of functors and distributors. In category theory, a profunctor is a functor from a product category to a category, generalizing the concept of a functor from a single category to a category. Profunctors are often used to describe the relationship between two categories, providing a way to associate a source category with a target category.

A profunctor is formally defined as a functor from the product category of two source categories to a target category, mapping each pair of objects from the source categories to an object in the target category. Profunctors can be thought of as a way to lift functors from the product category to the target category, enabling the computation of natural transformations between the lifted functors.

An important aspect of profunctors is their ability to describe adjunctions, a fundamental notion in category theory. An adjunction between two functors represents a pair of functors that are inversely related through a profunctor. Profunctors have practical applications in computer science, particularly in the study of differential calculus, denotational semantics, and program transformation.

The operations on profunctors are defined similarly to those of distributors, enabling the composition of profunctors. Profunctors also form an essentially algebraic category, meaning that their structure is determined by a set of algebraic equations. Profunctors play a key role in several fields, including automated reasoning, logical frameworks, and type theory.