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nLab > Latest Changes: double-negation topology

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  1.  
    • CommentRowNumber1.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 17th 2012
    • CommentRowNumber2.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 17th 2012

    We already had double negation.

    • CommentRowNumber3.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 17th 2012

    Okay. Suggestions for what to do?

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeAug 17th 2012

    I would prefer to put the topology stuff at double-negation topology, and if anything goes at double negation it should really only be something on double negation in logic. No time at the moment to do anything about it myself, I’m afraid.

    • CommentRowNumber5.
    • CommentAuthorTobyBartels
    • CommentTimeAug 17th 2012

    I’ve made the two pages link to each other. Urs has some structured material at double negation that you may want to put at double-negation topology.

    Personally, I don’t see how the pages are that different. The double-negation topology on a topos is logic. The other stuff at double negation is just decategorification and specialisation of that.

  6. I added the statement that the double-negation topology in a presheaf topos coincides with the dense topology to double negation.

    I also think that the creation of the two articles in question is just an accident and we should not keep both of them. I wouId title the page we keep ''double negation'' and keep its structure given by Urs; this subsumes the ''double negation topology''.

    If we would like to have an article which essentially is about the same (or closely related) topic but ''all topology'' we could call it ''dense topology''.
    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 19th 2012

    I also think that the creation of the two articles in question is just an accident and we should not keep both of them.

    While I don’t mind merging and adding redirects, I don’t understand the “should not”. What’s the harm, actually? I also don’t understand “keep its structure given by Urs” – it sounds like something fixed in stone.

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeAug 19th 2012

    In general, I think the harm in having almost-duplicate pages is that we end up writing almost-duplicate content, and wasting time on discussions like this one about what should go where. (-: Also someone may follow a link to one of the pages looking for something, and not realize that what they want is on the other page.

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 19th 2012

    I have incorporated the contents of what was double-negation topology into double negation, and then tried to eliminate the contents of the former article.

    I would like to have “double-negation topology” and all its kin redirect to double negation, and tried to do that, but I think I may have made a mess. I hope some kind Lab elf can help clean it up. Sorry for the extra work I’m making for people.

    I also think that the creation of the two articles in question is just an accident

    Certainly it was an accident. I was writing somewhere else and linked to an article named “double-negation topology”, confident that such an article already existed. After submitting, I found that was not the case (and obviously no redirect to double negation), so I went ahead and wrote up something short. I didn’t think to look up double negation, until David said something.

    • CommentRowNumber10.
    • CommentAuthorTobyBartels
    • CommentTimeAug 19th 2012
    • (edited Aug 19th 2012)

    Urs had made a redirect from double negation topology, but he neglected to do double-negation topology (or the plurals). (All of them work now.)

    I’ve cleaned up the mess. (The cache bug spared us, for once!)

    • CommentRowNumber11.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 19th 2012
    • (edited Aug 19th 2012)

    When I just hit double-negation topology, I didn’t get redirected to double negation.

    • CommentRowNumber12.
    • CommentAuthorTobyBartels
    • CommentTimeAug 19th 2012

    OK, that’s really weird, because it worked immediately after I made my comment above, but stopped working sometime in the next 4 minutes.

    So the cache bug takes a few minutes to appear !?!?!?

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeAug 20th 2012
    • (edited Aug 20th 2012)

    I also don’t understand “keep its structure given by Urs” – it sounds like something fixed in stone.

    Nobody said this, I think; or else I didn’t see it. In any case, it would certainly be weird. What I see is in #5 Toby saying that

    Urs has some structured material at double negation that you may want to put at double-negation topology.

    And I guess “structured material” just refers to the fact that it is sitting in a subsection and has Definition/Proposition-environments etc.

    • CommentRowNumber14.
    • CommentAuthorTobyBartels
    • CommentTimeAug 20th 2012

    Yes, that’s what I meant by ‘structured material’. I didn’t want that structure to be accidentally ignored, but it might be deliberately changed.

    • CommentRowNumber15.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 20th 2012

    @Urs: the phrase you quoted I quoted was from Stephan’s #6. While I didn’t fully understand it, I’m sure it was meant well.

    At this point I’m embarrassed not only by 15 comments on a mess I started, but also by the cache mess I made that seems hard to clean up. Maybe I need to learn, once and for all, exactly the correct steps to take to merge an article into another and effectively ’kill’ the prior article, in case this should come up again (although I thought I knew). Toby?

    • CommentRowNumber16.
    • CommentAuthorTobyBartels
    • CommentTimeAug 21st 2012
    • (edited Aug 21st 2012)

    To merge B into A:

    1. Edit A.
    2. Include [[!redirects B]] in A.
    3. Move B to B > history.
    4. Replace the contents of B > history with < [[B]].

    There is more detail in HowTo#merging.

    • CommentRowNumber17.
    • CommentAuthorTodd_Trimble
    • CommentTimeAug 21st 2012

    Okay, thanks! I thought I did steps 1-3 in that order, but I didn’t do 4. (Hm, for step 3 I changed the name of the page; maybe that’s not right. I’ll look at the page you linked.)

    • CommentRowNumber18.
    • CommentAuthorTobyBartels
    • CommentTimeAug 21st 2012

    Yes, changing the name is the same as moving. Somehow you created yet another page while doing that.

  19. Added to double negation the following example:

    Let 𝒪 X\mathcal{O}_X be the sheaf of continuous (or smooth, or holomorphic, or regular) functions on a topological space (or manifold, or complex manifold, or reduced scheme) XX. Then the pushforward of the pullback of 𝒪 X\mathcal{O}_X to the smallest dense sublocale of XX is the sheaf of rational functions on XX (i.e. sections over an open subset UU are given by sections of 𝒪 X\mathcal{O}_X defined on some dense open subset V⊆UV \subseteq U).

    • CommentRowNumber20.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 20th 2014

    I'd be inclined to say ‘meromorphic’ rather than ‘rational’. If 𝒪\mathcal{O} is the sheaf of polynomial functions (say for XX an affine space), then 𝒪 ¬¬\mathcal{O}_{\neg\neg} is not the sheaf of rational functions. But if 𝒪\mathcal{O} is the sheaf of holomorphic functions (for XX a complex manifold), then 𝒪 ¬¬\mathcal{O}_{\neg\neg} is the sheaf of meromorphic functions.

    But maybe this is what ‘rational’ means to an algebraic geometer?

  21. I don’t know the official convention in algebraic geometry, but you’re probably right: “meromorphic” has the connotation of “defined on some dense open subset”, while “rational” has the connotation of “expressible as the quotient of two functions”. I fixed the wording in the article.

    Note that for reduced schemes, the two notions agree in that the sheaf of meromorphic functions is canonically isomorphic to the total quotient ring of 𝒪 X\mathcal{O}_X. On non-reduced schemes, the two notions cannot agree in general, because the total quotient ring may then fail to be ¬¬\neg\neg-separated, i.e. there may be exist functions which agree on a dense open subset but do not agree on their whole domain.

    • CommentRowNumber22.
    • CommentAuthorTobyBartels
    • CommentTimeFeb 21st 2014

    Thanks, that's clear now!

    • CommentRowNumber23.
    • CommentAuthorThomas Holder
    • CommentTimeNov 30th 2014

    I’ve added some material at double negation. I am somewhat irritated as the entry contains the statement: The topos of double-negation sheaves is in fact the maximal Boolean subtopos contained in E which is hard to reconcile with a passage in Johnstone (1977) p.140 that claims that no such subtopos exists in general for elementary toposes.

    • CommentRowNumber24.
    • CommentAuthorMike Shulman
    • CommentTimeNov 30th 2014

    Without looking at it, I would guess that probably Johnstone is right and the entry is wrong (since that claim is a natural thing to expect/hope for). Does Johnstone give a counterexample?

    • CommentRowNumber25.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 30th 2014
    • (edited Nov 30th 2014)

    Is the trivial topos Boolean?

    EDIT: what does the Elephant say on the matter? I would take that over the 1977 book.

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2014

    The proposition in the entry explicitly speaks of sheaf toposes and gives a citation.

    (I am checking now, but it takes me a bit to get the book again…)

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2014
    • (edited Nov 30th 2014)

    Hm, I may have to wait until tomorrow with this.

    Incidentally

    says

    the subtopos of a given elementary topos consisting of its double negation sheaves can be seen as a universal way of making the topos Boolean, as it can be characterized as the largest dense Boolean subtopos of the given topos;

    and Mike here speaks of

    the double-negation-sheaves, which form the largest Boolean subtopos and the smallest dense subtopos.

    • CommentRowNumber28.
    • CommentAuthorMike Shulman
    • CommentTimeNov 30th 2014

    Don’t take my word for it; I could easily have been making a natural assumption that happens to be false. I am pretty sure that it is the smallest dense subtopos, but I could be wrong about the largest Boolean one.

    The trivial topos is Boolean. Of course, it is the smallest subtopos of any other topos.

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2014

    So I remember that I added that clause into the entry from a citation, and I remember speaking with a more professional topos theorist about it, too, but now I don’t find the citatation and I don’t really have the means and time resources either. If the statement is wrong of course it needs to be removed, but maybe one of you with five minutes and a better connection available (or else with a pen and the back of an envelope to produce a proof) could check.

    • CommentRowNumber30.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 1st 2014

    The double negation sheaves indeed form the smallest dense subtopos. I don’t have a precise reference for the proof, but Johnstone mentions this fact in the proof of A.4.5.21 in the Elephant.

    What is true is that ¬¬\neg\neg is the largest Lawvere-Tierney topology for which 0→10\to 1 is closed.

    Also, the topos of double negation sheaves is not trivial, if the original topos is not trivial. (which is what I was wondering in my question above about the trivial)

    • CommentRowNumber31.
    • CommentAuthorMike Shulman
    • CommentTimeDec 1st 2014

    Just looked at Johnstone ’77 and saw that he gives no example of a topos without a maximal Boolean subtopos. Can we come up with a simple one?

    • CommentRowNumber32.
    • CommentAuthorMike Shulman
    • CommentTimeDec 1st 2014

    BTW, evidently by “maximal” we actually mean “maximum” (i.e. the largest). (-:

    Here is I think an easy counterexample: the Sierpinski topos contains two obvious subtoposes each equivalent to SetSet and hence Boolean, but no proper subtopos containing both of them. So since it is not itself Boolean, it can have no largest Boolean subtopos. (In fact, one of those subtoposes is double-negation, while the other is disjoint from it.)

    • CommentRowNumber33.
    • CommentAuthorThomas Holder
    • CommentTimeDec 1st 2014
    • (edited Dec 1st 2014)

    ad #29: the elephant actually gives a short proof of the characterization as largest dense topology right before A.4.5.21 at the bottom of page 219.

    • CommentRowNumber34.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2014

    Mike, re #32: I suppose only one these is dense, though?

    • CommentRowNumber35.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2014

    I feel a bit begind with this discussion now. First of all I have started now a stub (dense,closed)-factorization. Needs more, please add to it.

    • CommentRowNumber36.
    • CommentAuthorMike Shulman
    • CommentTimeDec 1st 2014

    Re 34, yes, only the double-negation one is dense. That is, the closed point in the Sierpinski space is dense, but the open point isn’t.

    • CommentRowNumber37.
    • CommentAuthorZhen Lin
    • CommentTimeDec 1st 2014

    Surely, it’s the open point that is dense and the closed one that is not?

    • CommentRowNumber38.
    • CommentAuthorMike Shulman
    • CommentTimeDec 2nd 2014

    Yes, of course, sorry, brain turned off.

1 to 38 of 38

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