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

[Derek Elkins]

On Tue, 2008-03-04 at 18:30 +0000, 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..

I explicitly mentioned this at the end of my first email:

"""
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.
"""

I'm suggesting that trying to find such an analogy may be more trouble
than it is worth.   The best analogy I've heard is to relate the problem
of finding an adjunction to optimization problems.  Personally, I find
representability to be simpler, more useful, and easier to get an
intuition about.  Adjunction is then a particular case of parameterized
representability.