On Tue, 2008-03-04 at 17:16 +0000, Edsko de Vries wrote:
Hi,
Is there an intuition that can be used to explain adjunctions to functional programmers, even if the match isn't necessary 100% perfect (like natural transformations and polymorphic functions?).
Well when you pretend Hask is Set, many things can be discussed fairly directly. F is left adjoint to U, F -| U, if Hom(FA,B) is naturally isomorphic to Hom(A,UB), natural in A and B. A natural transformation over Set is just a polymorphic function. So we have an adjunction if we have two functions: phi :: (F a -> b) -> (a -> U b) and phiInv :: (a -> U b) -> (F a -> b) such that phi . phiInv = id and phiInv . phi = id and F and U are instances of Functor. The unit and counit respectively is then just phi id and phiInv id. You can work several examples using this framework, though it gets difficult when it is difficult to model other categories. Also discussing and proving some properties of adjunctions (namely continuity properties) would help. Of course, this is a concrete example using basic ideas of programming and not some "intuitive analogy". I feel comfortable working with adjunctions, but I don't have some general analogy that I use.