From: Paul Levy <pbl@dcs.qmw.ac.uk>
To: bhalchin@hotmail.com
Cc: w.james@latrobe.edu.au, categories@mta.ca
Subject: Re: co-exponential question
Date: Tue, 20 Jul 1999 18:41:59 +0100 (BST) [thread overview]
Message-ID: <199907201741.SAA29163@gentzen.dcs.qmw.ac.uk> (raw)
In-Reply-To: <3794E2AC.425B@latrobe.edu.au> (message from William James on Tue, 20 Jul 1999 13:57:18 -0700)
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/
===========================================================================
next prev parent reply other threads:[~1999-07-20 17:41 UTC|newest]
Thread overview: 5+ messages / expand[flat|nested] mbox.gz Atom feed top
1999-07-16 19:55 Bill Halchin
1999-07-20 20:57 ` William James
1999-07-20 17:41 ` Paul Levy [this message]
1999-07-22 20:28 Andrzej Filinski
1999-07-23 19:40 ` Peter Selinger
Reply instructions:
You may reply publicly to this message via plain-text email
using any one of the following methods:
* Save the following mbox file, import it into your mail client,
and reply-to-all from there: mbox
Avoid top-posting and favor interleaved quoting:
https://en.wikipedia.org/wiki/Posting_style#Interleaved_style
* Reply using the --to, --cc, and --in-reply-to
switches of git-send-email(1):
git send-email \
--in-reply-to=199907201741.SAA29163@gentzen.dcs.qmw.ac.uk \
--to=pbl@dcs.qmw.ac.uk \
--cc=bhalchin@hotmail.com \
--cc=categories@mta.ca \
--cc=w.james@latrobe.edu.au \
/path/to/YOUR_REPLY
https://kernel.org/pub/software/scm/git/docs/git-send-email.html
* If your mail client supports setting the In-Reply-To header
via mailto: links, try the mailto: link
Be sure your reply has a Subject: header at the top and a blank line
before the message body.
This is a public inbox, see mirroring instructions
for how to clone and mirror all data and code used for this inbox;
as well as URLs for NNTP newsgroup(s).