Well, we have at least one very useful example of adjunction. It's called "curry". See, if X is some arbitrary type, you can define type F = (,X) instance Functor F where fmap f (a,x) = (fa,x) type G = (->) X instance Functor G where fmap f h = \x -> f (h x) Now, we have the adjunction: phi :: ((a,X) -> b) -> (a -> (X -> b)) phi = curry phiInv :: (a -> (X -> b)) -> ((a,X) -> b) phiInv = uncurry On 4 Mar 2008, at 21:30, Edsko de Vries wrote:
On Tue, Mar 04, 2008 at 11:58:38AM -0600, Derek Elkins wrote:
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.
Sure, but that doesn't really explain what an adjunction *is*. For me, it helps to think of a natural transformation as a polymorphic function: it gives me an intuition about what a natural transformation is. Specializing the definition of an adjunction for Hask (or Set) helps understanding the *definitions*, but not the *intention*, if that makes any sense..
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