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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeJun 30th 2019
   • PermaLink
   Author: Mike Shulman
   Format: MarkdownItex(Made a trivial edit in order to create the discussion page.) I don't like this terminology. In all other contexts I can think of, denseness is a *property* of a subset or subtopos. This particular subtopos happens to be a dense one, and the dense sieves for this topology are called by set-theorists merely "dense" because this is the only topology on posets they consider. But it's not the unique dense topology on a category, so we shouldn't call it "the" dense topology. Why not call it the "double-negation topology" since that's what it is? <a href="https://ncatlab.org/nlab/revision/diff/dense+topology/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/dense+topology/10">v10</a>, <a href="https://ncatlab.org/nlab/show/dense+topology">current</a>

   (Made a trivial edit in order to create the discussion page.)

   I don’t like this terminology. In all other contexts I can think of, denseness is a property of a subset or subtopos. This particular subtopos happens to be a dense one, and the dense sieves for this topology are called by set-theorists merely “dense” because this is the only topology on posets they consider. But it’s not the unique dense topology on a category, so we shouldn’t call it “the” dense topology. Why not call it the “double-negation topology” since that’s what it is?

   diff, v10, current

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJul 6th 2019
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   Author: Mike Shulman
   Format: MarkdownItexIf no one objects, I may rename this page to [[double-negation topology]] and move the topological material from [[double negation]] over here. (A while ago we had a discussion about duplication between the then-existing page [[double-negation topology]] and the page [[double negation]], which was resolved by merging them. But I think in light of the fact that this page also exists, it's worthwhile re-separating them and instead resolving the potential duplication by restricting the page [[double negation]] to talk mainly about the *operation* of double negation, with the topological aspects restricted mainly to their dedicated page.)

   If no one objects, I may rename this page to double-negation topology and move the topological material from double negation over here.

   (A while ago we had a discussion about duplication between the then-existing page double-negation topology and the page double negation, which was resolved by merging them. But I think in light of the fact that this page also exists, it’s worthwhile re-separating them and instead resolving the potential duplication by restricting the page double negation to talk mainly about the operation of double negation, with the topological aspects restricted mainly to their dedicated page.)

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 6th 2019
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   Author: Dmitri Pavlov
   Format: MarkdownItexMac Lane and Moerdijk refer to this topology both as a "dense topology" and as a "¬¬-topology" (III.2(e), page 115 in their book).

   Mac Lane and Moerdijk refer to this topology both as a “dense topology” and as a “¬¬-topology” (III.2(e), page 115 in their book).

4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeJul 6th 2019
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   Author: Mike Shulman
   Format: MarkdownItexI don't think I've seen the term "dense topology" anywhere else; have you?

   I don’t think I’ve seen the term “dense topology” anywhere else; have you?

5. • CommentRowNumber5.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 6th 2019
   • (edited Jul 6th 2019)
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   Author: Dmitri Pavlov
   Format: MarkdownItexBorceux (Handbook of Categorical Algebra, Volume 3, Section 9.4) talks about double negation sheaves and double negation topology. Johnstone (Elephant, A.4.5.9, A.4.5.21) simply talks about the “smallest dense subtopos”. So the three (arguably) main book-length sources use three different names for this notion. I guess this means there is no established terminology and we should mention all three names. But “dense topology” does sound weird.

   Borceux (Handbook of Categorical Algebra, Volume 3, Section 9.4) talks about double negation sheaves and double negation topology.

   Johnstone (Elephant, A.4.5.9, A.4.5.21) simply talks about the “smallest dense subtopos”.

   So the three (arguably) main book-length sources use three different names for this notion. I guess this means there is no established terminology and we should mention all three names. But “dense topology” does sound weird.

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeJul 7th 2019
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   Author: Mike Shulman
   Format: MarkdownItexI think double-negation is clearly the winner by majority vote of those three: in addition to Borceaux using it, Mac Lane and Moerdijk at least mention it, and Johnstone does use $\neg\neg$ as *notation* for the local operator even if he doesn't write out "double-negation" or "$\neg\neg$-topology" in English.

   I think double-negation is clearly the winner by majority vote of those three: in addition to Borceaux using it, Mac Lane and Moerdijk at least mention it, and Johnstone does use $\neg\neg$ as notation for the local operator even if he doesn’t write out “double-negation” or “$\neg\neg$-topology” in English.

7. • CommentRowNumber7.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 7th 2019
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   Author: Dmitri Pavlov
   Format: MarkdownItexIt's hard to disagree. “Double negation topology” is much more clear and specific than “dense topology” anyway.

   It’s hard to disagree. “Double negation topology” is much more clear and specific than “dense topology” anyway.