The Monad Called Free

The article by romes delves into the conceptual underpinnings of the Free monad in Haskell, positioning it as a 'higher-order monad' and drawing connections to category theory. The author aims to fill a perceived gap by implementing Free as an instance of a higher-order monad type class, borrowing from Ed Kmett's category-extras.

• The piece begins by defining the standard Free monad data type and its Functor and Monad instances.
• It introduces Endo, the category of endofunctors, where objects are Hask Functors and arrows are Natural transformations.
• A HFunctor type class is defined for functors operating within Endo, equipped with ffmap and hfmap.
• The author then draws direct analogies between standard Hask constructions (like products, sums, and lists) and their counterparts in Endo.
• A key insight is presented: Free monads can be understood as the Endo-flavored equivalent of lists, and more generally, as free monoids in the category of endofunctors.
• This analogy is further developed by generalizing common list operations such as singleton and foldMap to hsingleton and hFoldMap for Free monads.
• The article demonstrates how hFoldMap can interpret programs represented by Free monads acting as abstract syntax trees.
• Finally, the Free monad is made an instance of a proposed HMonad type class, fulfilling the initial goal of treating Free as a higher-order monad.

The article concludes by highlighting the profound and non-trivial similarities between these abstract categorical constructs and practical functional programming patterns. It offers a sophisticated view of how Free monads unify concepts across different levels of abstraction in functional programming.