popl2008 ** 1. GADTs are syntactic sugar. The real structures at work are existentially quantified types and the equality GADT. 2. Indexing by a set, rather than a category, as GADTs do means that really this is dependently typed programming faked in Haskell by using kind level proxies for types 3. As a result, Haskell ends up with a curious "logic programming" feel to its type level computation and functional programming feel to its term-level computation. PJohann and NGhani ** his main point (which worked with me) is that we should pay more attention Kan extensions. We already know and love (strong) functors, (co)monads, adjunctions and the Yoneda lemma.
Metamorphic programming is an approach to extend the structured recursive programming discipline, which favors the use of fold operations over general recursion, to abstract data types. The key idea is to represent an ADT by two parts, a constructorand a destructor,which are essentially functions to/from a common representation. Then a fold can work on an ADT by applying parameter functions to values that are delivered by the ADT's own destructor. Fold operations that use as a parameter the constructor of another ADT, called ADT transformers,play an important role and offer a concise programming style. Several laws for ADT folds and transformers exist that can be used for program optimization and verification.
An introduction to 2-categories is given by illustrating how the structure of typed lambda calculus may naturally be viewed as a 2-category. In this vein. the structure of computations or conversions gives rise to notions of lax 2':"adjointness.