The boringness of the dual of exponential
Dear all, 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? [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Sat, 5 Nov 2011 12:52:03 +0000, David Leduc wrote:
... 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?
Is there a definition behind this conjecture? If so, TIA. Cheers, -- Fred [PS: this also tests submission of posts via Gmane, aka news.gmane.org .] [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
The lattice of all subtoposes of any given toposis surely not boring.For example the lattice of positive model classes of a given theory needs investigating. Bill
Date: Sun, 6 Nov 2011 22:55:08 +0100 From: streicher@mathematik.tu-darmstadt.de To: david.leduc6@googlemail.com CC: categories@mta.ca Subject: categories: Re: The boringness of the dual of exponential
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
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 05/11/2011, at 11:52 PM, David Leduc wrote:
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?
The conjecture is false. Take any category E where exponentiable is interesting. Then the dual of exponentiable is not boring in E^op. ==Ross [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Perhaps you are thinking of dualizing part of the notion of exponential, but not the other part? http://ncatlab.org/nlab/show/cocartesian+closed+category Mike On Sat, Nov 5, 2011 at 05:52, David Leduc <david.leduc6@googlemail.com> wrote:
Dear all,
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?
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (6)
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David Leduc -
F. William Lawvere -
FEJ Linton -
Michael Shulman -
Ross Street -
Thomas Streicher