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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 16th 2011
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded to [[double negation topology]] some of the basic statements (in the section on topos theory)

   added to double negation topology some of the basic statements (in the section on topos theory)

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeMar 11th 2015
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   Author: Mike Shulman
   Format: MarkdownItexAdded to [[double negation]] a proof that it is the unique topology which is both dense and Boolean.

   Added to double negation a proof that it is the unique topology which is both dense and Boolean.

3. • CommentRowNumber3.
   • CommentAuthorThomas Holder
   • CommentTimeMar 26th 2015
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   Author: Thomas Holder
   Format: MarkdownItexI've added some consequences of the interaction between $\neg\neg$ and Aufhebung of $\emptyset\dashv\ast$ at [[Aufhebung]]. This might profit from some expert's reality check though.

   I’ve added some consequences of the interaction between and Aufhebung of at Aufhebung. This might profit from some expert’s reality check though.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMar 27th 2015
   • PermaLink
   Author: Urs
   Format: MarkdownItexThomas, in your [remark in the entry](http://ncatlab.org/nlab/show/Aufhebung#OnDoubleNegation), why does it follow that > in particular $\mathcal{E}_{\neg\neg}=\mathcal{E}_j$ in case the former is [[essential geometric morphism|essential]] ?

   Thomas, in your remark in the entry, why does it follow that

   in particular in case the former is essential

   ?

5. • CommentRowNumber5.
   • CommentAuthorThomas Holder
   • CommentTimeMar 27th 2015
   • PermaLink
   Author: Thomas Holder
   Format: MarkdownItex$\mathcal{E}_j$, the Aufhebung of $\emptyset\dashv\ast$ (which by definition has to be essential - a [[level]]), as a dense subtopos is always $\mathcal{E}_{\neg\neg}\subseteq\mathcal{E}_j$ by general facts on [[double negation topology]], so provided $\mathcal{E}_{\neg\neg}$ _is_ a level, it always wins out over other (dense) candidates. Lawvere discusses essentiality of $\neg\neg$ as a possible axiom in the Como 1991 paper and $\mathcal{E}_{\neg\neg}\subseteq\mathcal{E}_j$ occurs in the 1989 'taco' paper. What slightly surprises me is that $\mathcal{E}_{\neg\neg}=\mathcal{E}_j$ holds for all cohesive sites and that $\mathcal{E}_{\neg\neg}$ is the only possible candidate for _Boolean_ Aufhebung of $\emptyset\dashv\ast$.

   , the Aufhebung of (which by definition has to be essential - a level), as a dense subtopos is always by general facts on double negation topology, so provided is a level, it always wins out over other (dense) candidates. Lawvere discusses essentiality of as a possible axiom in the Como 1991 paper and occurs in the 1989 ’taco’ paper. What slightly surprises me is that holds for all cohesive sites and that is the only possible candidate for Boolean Aufhebung of .

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeMar 27th 2015
   • (edited Mar 27th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, now I see what you mean. I have taken the liberty of slightly rephrasing that part in the remark to make it parse unambiguously.

   Thanks, now I see what you mean. I have taken the liberty of slightly rephrasing that part in the remark to make it parse unambiguously.

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMar 28th 2015
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have also added pointer to p. 8 in Lawvere91 to the relevant [remark here](http://ncatlab.org/nlab/show/double+negation#RelationToAufhebungof01) at _[[double negation]]_.

   I have also added pointer to p. 8 in Lawvere91 to the relevant remark here at double negation.