Tell me about Polymorphic Lambda Calculus, who invented it, and how is it being used? Is it related to Polynomial Functors? [Polymorphic lambda calculus](https://en.wikipedia.org/wiki/Polymorphic_lambda_calculus ) , a.k.a. [[System F]], is an extension of the [[lambda calculus]], a formal system for expressing computation. It allows for the abstraction and manipulation of functions with polymorphic types, meaning functions that can operate on values of different types. The concept of polymorphic lambda calculus was first introduced by [[John C. Reynolds]] in 1974 as part of his work on type systems. He developed the [[System F]], also known as the second-order lambda calculus, which includes polymorphic types and quantification over type variables. Polymorphic lambda calculus has found applications in various areas of computer science. It forms the basis for typed functional programming languages like Haskell, ML (Meta Language), and others that support parametric polymorphism. These languages leverage polymorphism to write reusable code and enable type-safe generic programming. # Connections to Polynomial Functors In relation to polynomial functors, there is a connection between them through the concept of parametricity. Parametricity is a principle that describes how functions must behave based solely on their types. Polymorphic lambda calculus provides a way to reason about parametricity, and polynomial functors are one means to capture this behavior precisely. Polynomial functors are used in functional programming to model recursive data structures like lists, trees, and more complex structures. They provide a compositional approach to defining these structures by combining simpler ones using specific operations called functorial mappings. The connection to polymorphic lambda calculus arises from the ability to express these mappings using higher-order functions in a type-safe manner. # Conclusion Overall, while there is a relationship between polymorphic lambda calculus and polynomial functors through their shared focus on parametricity and type-safe programming, they are distinct concepts that serve different purposes within functional programming paradigms.