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
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