[Haskell-cafe] Join and it's relation to >>= and return

[Frank Atanassow]

On Jun 9, 2004, at 9:39 AM, Jerzy Karczmarczuk wrote:
>
> I have *nothing* to add, just a question.
> Do you /anybody/ know of any edible work on ADJUNCTIONS in the context 
> of Haskell structures? Perhaps instead of searching for 'inverses' one 
> should think more about adjoints?...

Yes, I think this is the right way to go. If you look at work by Power, 
Thielecke and Streicher on continuations [*], you will find that they 
model type negation as a self-adjoint functor on a closed premonoidal 
category, and IIRC a closed premonoidal category is equivalent to a 
thing called a closed kappa-category with a computational monad on it. 
The self-adjointness corresponds to the involutivity of negation.

This all depends on the tensor product being commutative. It's also 
possible to drop commutativity, in which case you end up with _two_ 
exponentials, also called residuals, corresponding to the two ways to 
curry a non-commutative product. Such a category can serve as a 
proof-theoretic model for Lambek calculus, which is used in categorial 
grammar.

In this situation, if you demand a multiplicative inverse, or 
equivalently a dual to the unit, you get _two_ negations or (better I 
think) `dualizers'. Barr treats this in a few of his papers, including 
[1], and [2] for the simpler commutative case. There is a neat way of 
forming such categories, called the Chu construction [2], which has 
provoked much interest in the categorical community.

I once wrote some notes on something I called action logic, which tried 
to organize these ideas at a simpler level. The logic was 
non-commutative and had two dualizers like I said, and was designed to 
be sound and complete for a model where every proposition was 
interpreted by an adjoint pair of functions (or, just a Galois 
connection), and dualization by replacing a function by its (unique) 
adjoint. It all worked out very nicely because adjoints have such neat 
properties. I still have the notes if you're interested.

[*] CiteSeer seems kind of broken, so try:

http://www.google.com/search?q=site%3Aciteseer.ist.psu.edu+thielecke

[1] Michael Barr, Non-symmetric *-autonomous categories, 1999.
     ftp://ftp.math.mcgill.ca/pub/barr/asymmps.zip

[2] http://citeseer.ist.psu.edu/251562.html

[3] http://citeseer.ist.psu.edu/barr96chu.html

Regards,
Frank