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From: Paul Levy <pbl@dcs.qmw.ac.uk>
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Subject: Re: co-exponential question
Date: Tue, 20 Jul 1999 18:41:59 +0100 (BST)
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I don't know if this is relevant to your question, but there is an example in
programming semantics where the dual of a cartesian closed
category has independent significance, due to Lafont, Streicher, Reus
and Hofmann.  (Some further work was done by Selinger.)  It is found in the following paper:
		  
@Article{StreicherReus:continuations,
  title =        "Classical logic, continutation semantics and abstract
                 machines",
  author =       "Th. Streicher and B. Reus",
  pages =        "543--572",
  journal =      "Journal of Functional Programming",
  month =        nov,
  year =         "1998",
  volume =       "8",
  number =       "6",
}

The construction is as follows.

Let C be a distributive
category, and let Ans (the "answer type") be an object in C, such that the exponential X -> Ans exists for
each object X.  Define two categories K and
N as follows. Both have the same objects as C.  In K, a morphism from
X to Y is a C-morphism from X x (Y -> Ans) to Ans.  In N, a morphism
from X to Y is a C-morphism from (X -> Ans) x Y to Ans. 

 Clearly these two
categories are dual.  Streicher and Reus' paper makes it clear that just as K can be
used to interpret a typed call-by-value language with control effects (which
was well-known), N can be used to interpret a typed call-by-name language
with control effects.  Like any call-by-name model, N is cartesian closed:

the product of X and Y in N is given by X+Y

the exponential from X to Y in N is given by (X -> Ans) x Y

K is certainly an important category, but I wouldn't say that the fact
that it has coexponentials is significant.  

Paul

===========================================================================
Paul Blain Levy, Department of Computer Science,
Queen Mary and Westfield College, LONDON E1 4NS
http://www.dcs.qmw.ac.uk/~pbl/
===========================================================================