[Haskell-cafe] Relevance and applicability of category theory
Aaron Altman
aaltman at pdx.edu
Wed Jan 30 21:33:26 EST 2008
David's post looks like it has cleared up what I was wondering about. I
think it was the specific contents of Ob(Hask) and Haskell functions as
arrows in Hask that I was unclear on. With that in mind I understand
and agree with Miguel's explanations. Thanks, guys.
Dan Weston wrote:
> I assume you have read the references in
> http://www.haskell.org/haskellwiki/Research_papers/Monads_and_arrows
>
> The penultimate sentence in question 1 below (role of adjoint functors
> in Haskell monads) was addressed by David Menendez in a recent post to
> this list:
>
> http://www.haskell.org/pipermail/haskell-cafe/2007-December/036361.html
>
> There was discussion of Category theory monad <----> Haskell monad
> subsequent to the thread
>
> http://www.haskell.org/pipermail/haskell-cafe/2005-August/011021.html
>
> A categorical approach to graph transformations with Haskell is
> described in:
>
> http://philippsen.com/Personen/schneide/gtbook/appendix-a.pdf?language=de
>
> There are several Haskell modules (named e.g. Category) floating
> around that attempt to better reflect categorical usage in Haskell,
> but I do not have a reference handy.
>
> Dan
>
> aaltman at pdx.edu wrote:
>> Category theory seems to have an inconsistent relationship to Haskell
>> - both documentation and the language's implementations of
>> categorical concepts. I come from a math background that makes the
>> Haskell's tight coupling to its mathematical foundation very
>> appealing. But, I may have found some inconsistencies. Whether you
>> agree or not I hope we can get some discussion going that will
>> clarify the wikibooks, etc.
>>
>> My issues:
>> 1. Are Haskell monads useful in a truly categorical sense?
>> 2. Is Haskell's functor class misnamed?
>> 3. Haskell arrows and Haskell monads have a misleading relationship
>>
>> 1. Categorical monads are a class of functors, that is, morphisms on
>> Cat. Haskell's monads are at least a bit closer to the categorical
>> idea than Haskell's functors by virtue of having the same domain and
>> codomain: Hask -> Hask. I think applicability of Haskell monads
>> beyond sequencing computation, and the validity of their definition,
>> would be much more clear if someone explained the meaning of adjoint
>> functors to and from Hask. In other words, provide the mathematical
>> characterization to make Haskell monads a precise representation of
>> categorical monads on Hask.
>>
>> 2. Functors are structure preserving maps in the category Cat. The
>> Haskell Functor class represents structure preserving maps in the
>> category Hask, which seem to me more like the categorical notion of
>> arrows or the algebraic notion of homomorphisms. If we talk about
>> functors relating to Haskell, it seems more likely we'd be referring
>> to functors to or from the category Hask than functions (or arrows or
>> whatever) between elements of Hask. Maybe categorical functors are
>> useful as an abstract characterization of programs themselves: a
>> program defines a functor from the problem domain into Hask, or from
>> Hask into output in a problem domain. The name "Arrow" is already
>> taken; what about renaming Functor to Automorphism?
>>
>> 3. I believe the documentation stating that Haskell arrows are a
>> generalization of Haskell monads, but arrows are a categorical thing
>> too and in that context bear a much more distant relationship to
>> monads. Does a Haskell arrow have Hask as domain and codomain? Or
>> is one particular element in Hask its domain and possibly another its
>> codomain? Those are not at all the same thing.
>>
>> My guess is that the concept of Hask is too murky to be any use right
>> now. The documentation may be more clear by avoiding it until we
>> work out a formal definition... or is there one already?
>>
>> - Aaron Altman
>>
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