[Haskell-cafe] Relevance and applicability of category theory

[aaltman at pdx.edu]

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