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..
Edsko
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