Kan Extensions in Double Categories

This article delves into the theory of Kan extensions within the framework of double categories, specifically proarrow equipments. It generalizes the traditional notion of Kan extensions by replacing representable profunctors with general profunctors, allowing for bending vertical arrows using horizontal conjoints. The article provides a detailed categorical formulation of right Kan extensions, including the universal condition and factorization properties, and translates these concepts into Haskell code using the Ran data type. It also discusses the computational meaning and extends the generalization to left Kan extensions. The content is highly technical, focusing on category theory, profunctors, and functional programming, making it relevant to advanced programmers and computer scientists interested in abstract algebra and type theory.