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@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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