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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
   • PermaLink
   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 $\neg\neg$ and Aufhebung of $\emptyset\dashv\ast$ 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 $\mathcal{E}_{\neg\neg}=\mathcal{E}_j$ 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 $\emptyset\dashv\ast$.

   $\mathcal{E}_j$, 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 $\emptyset\dashv\ast$.

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.