Re: Logic preserved in double negation subtopos?

["Prof. Peter Johnstone" <P.T.Johnstone@dpmms.cam.ac.uk>]

The inclusion of double-negation sheaves is an example of what I
called a sub-open map in my paper "Open maps of toposes"
(Manuscripta Math. 31 (1980), 217-247). Sub-open maps have the
property that their inverse image functors commute with implication
-- indeed, one could take that as a definition, although it
wasn't how I defined them in the paper.

Peter Johnstone
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On Tue, 2 Sep 2003, Jonas Eliasson wrote:

> While writing a joint paper with Steve Awodey, we came to think about the
> following question:
>
> Given a Grothendieck topos Sh(C), what logic is preserved by the
> associated sheaf functor from Sh(C) to the double negation subtopos of
> Sh(C)?
>
> We know that a: Sh(C) --> DNSh(C) preserves geometric logic. Since it is
> double negation it also preserves 0 (falsehood), negation and implication.
> >From this you can draw the conclusion that a preserves the validity of
> formulas built up from double negation stable predicates without universal
> quantifiers.
>
> Presumably this has been studied in the literature, can something stronger
> be said about what validities are preserved, could anyone provide a
> reference for a general result of this kind?
>
> Grateful for any help,
> Jonas Eliasson
>
>
>
>
>  ------------------------------------------
> | Jonas Eliasson                           |
> | Department of Mathematics                |
> | Uppsala University                       |
> | Sweden                                   |
> | E-mail: jonase@math.uu.se                |
> | Homepage: http://www.math.uu.se/~jonase/ |
>  ------------------------------------------
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