[Haskell-cafe] Functional programmer's intuition for adjunctions?

[Edsko de Vries]

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