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From: Thomas Streicher <streicher@mathematik.tu-darmstadt.de>
To: David Leduc <david.leduc6@googlemail.com>
Cc: categories <categories@mta.ca>
Subject: Re: The boringness of the dual of exponential
Date: Sun, 6 Nov 2011 22:55:08 +0100	[thread overview]
Message-ID: <E1RNBk4-0007xD-TH@mlist.mta.ca> (raw)
In-Reply-To: <E1RN2tu-0005v9-2T@mlist.mta.ca>

> I have a conjecture: the dual of exponential is boring in any
> category. Is there a counterexample to this conjecture? If not how
> can we prove it?

Well, if boring means non-trivial there are examples, namely opposites
of cartesian closed categories. E.g. Set^op equivalent to CABA
(complete atomic boolean algebras). E.g. for a topological space X the
poset C(X) of closed subsets of X ordered by set inclusion is an example.
There conegation ~A is the closure of the complement of A and A \cap ~A
is the border of A. This received some attention as models of
"dialectical logic". There are also biHeyting algebras meaning that A
and A^op are Heyting algebras. Subobject lattices of objects in
presheaf toposes are examples of this as observed by Lawvere.
But if you mean by coexponential A x (_) having a left adjoint then
in presence of a terminal object 1 this means A is isomorphic to 1 and
indeed they are trivial.

Thomas


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  parent reply	other threads:[~2011-11-06 21:55 UTC|newest]

Thread overview: 22+ messages / expand[flat|nested]  mbox.gz  Atom feed  top
2011-11-05 12:52 David Leduc
2011-11-06 20:22 ` FEJ Linton
2011-11-06 21:55 ` Thomas Streicher [this message]
2011-11-07 16:32   ` F. William Lawvere
2011-11-06 22:59 ` Ross Street
     [not found] ` <F284B070-BBE5-4187-BA3C-E1A3EA560E6A@mq.edu.au>
2011-11-07 12:52   ` David Leduc
2011-11-08 16:20     ` Paul Taylor
2011-11-09 20:57       ` Uwe.Wolter
2011-11-10  9:29       ` Prof. Peter Johnstone
2011-11-11  7:47         ` Vaughan Pratt
2011-11-11 21:08           ` Robert Seely
2011-11-09 11:28     ` Andrej Bauer
2011-11-10  0:45       ` Jocelyn Ireson-Paine
2011-11-13  7:57         ` Vaughan Pratt
2011-11-14 13:36           ` Patrik Eklund
2011-11-15 13:03             ` Robert Dawson
     [not found]               ` <07D33522-CA8F-4133-A8E8-4B3BF6DFCCB4@cs.ox.ac.uk>
2011-11-16 18:06                 ` Robert Dawson
2011-11-10  2:17     ` Peter Selinger
2011-11-07 21:23 ` Michael Shulman
2011-11-10  1:11 ` Andrej Bauer
2011-11-09  9:19 Reinhard Boerger
2011-11-09 18:58 RJ Wood

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