[Haskell-cafe] Relevance and applicability of category theory

[Dan Weston]

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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