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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 9th 2019

    Richards asks that we report bugs in a single thread, and if discussion is needed start a new thread for each.

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 9th 2019

    We have the issue of the link !-modality not rendering. Maybe because of the punctuation at the front. This was discussed here.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 9th 2019

    Then there’s the issue of consecutive links not always generating a space between them.

  1. Thank you David! I will use this thread as the main place for keeping track of things from now. If it’s not here I may miss it :-).

    • CommentRowNumber5.
    • CommentAuthorRichard Williamson
    • CommentTimeJul 10th 2019
    • (edited Jul 10th 2019)

    Investigate use of max-width in mathematics CSS. Consider embedding DejaVu fonts. See here.

    • CommentRowNumber6.
    • CommentAuthorAlexisHazell
    • CommentTimeJul 10th 2019

    Re. #2, my guess is that the problem is that ! is used within Wiki-links by the !include and !redirects commands, so a page beginning with ! confuses the parser into thinking it’s dealing with something other than a page name.

    • CommentRowNumber7.
    • CommentAuthoratmacen
    • CommentTimeJul 10th 2019

    I had a problem with itex negative spacing. Thread here.

    • CommentRowNumber8.
    • CommentAuthoratmacen
    • CommentTimeJul 10th 2019

    There seemed to be an SVG problem here.

    • CommentRowNumber9.
    • CommentAuthorAlexisHazell
    • CommentTimeAug 26th 2019

    The “need to make an additional edit for uploaded file link to appear” issue is not new, but for tracking purposes, i’ve added it to the Technical TODO list.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJan 13th 2020
    • (edited Jan 13th 2020)

    Just to highlight that there is a problem with the rendering of edit announcements by user Ben in the thread on bicategories here

    [edit: ah, seeing now that also my link above doesn’t render!]

    • CommentRowNumber11.
    • CommentAuthoratmacen
    • CommentTimeJan 13th 2020

    All the comments are getting interpreted as plain text. Weird. Does that happen to me too?

    • CommentRowNumber12.
    • CommentAuthoratmacen
    • CommentTimeJan 13th 2020

    My comment looks good to me.

  2. The reason for #10 not originally rendering was that ’Text’ was selected as the format type. I have corrected this now. I suspect this was simply a case of Urs in a rush.

    The comments in the other thread seem also to have been rendered in ’Text’ format, but I do not immediately see how that has happened. I will look some more.

  3. Aha! It seems that the nForum ’Ben’ user is banned. This must cause comments to be rendered as ’Text’. I suspect the banned ’Ben’ is not the same as the nLab one, so I will try to fix this by deleting the old, banned Ben.

  4. Fixed now by deleting the old, banned Ben and re-assigning the authorship of a few comments.

    • CommentRowNumber16.
    • CommentAuthorPaoloPerrone
    • CommentTimeJan 15th 2020

    The page algebra over a monad seems impossible to edit: trying gives “500 internal server error”.

    • CommentRowNumber17.
    • CommentAuthorAlexisHazell
    • CommentTimeJan 15th 2020

    @PaoloPerrone, re. #16: I’m currently looking into this; thanks for reporting!

  5. What happened in this case is that the page failed to be fully created. More precisely, for those familiar with the internal details, a row was created in the table ’pages’ but not in the table ’revisions’. This led also to the 500 error when carrying out a search.

    I have now deleted the page, and re-created it: algebra over a monad. I did not have your attempted content available in an easily accessible form, Paulo (I have now tweaked the code so we would have the source available on a future occasion), so I must ask you to edit the page to put in the content. Fortunately, I believe you mentioned that you have a backup.

    I will see if I can fix the root cause of this problem when I get a chance.

  6. The page Critique of Pure Reason is impossible to edit giving the error:

    An unexpected error occurred when rendering the page
    
    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeJan 19th 2020

    On a different note:

    As of this morning, I find that hyperlinks coded as

      [arXiv:hep-th/0206192](https://arxiv.org/abs/hep-th/0206192)
    

    do not render properly on the nLab (it works here on the nForum!)

    In fact, on the nLab also the escaped form as above does not render properly: the “](” bit gets removed, and that then also explains the bad rendering of the un-escaped form.

    I have put an example with the problem in the Sandbox.

  7. Re #20: Fixed now. See here.

  8. Re #19: It seems possible to edit now, I guess it was probably the same root cause as that which led to #20.

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeJan 19th 2020

    Thanks a million! Great.

    • CommentRowNumber24.
    • CommentAuthorPaoloPerrone
    • CommentTimeJan 19th 2020

    @Richard: Okay, thank you. I’ll update the content now.

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeMar 10th 2020
    • (edited Mar 10th 2020)

    Is there a way to add captions to floating images?

    As in the example here, consider a bunch of paragraphs, each flowing around a respective floating image.

    Then how to put the source reference for the image (“graphics taken from XYZ”) right next to (below) the floating image, instead of it flowing with the other text?

    Is this possible with the existing imagefromfile-functionality?

    If not, I am requesting such a feature hereby! :-)

  9. Yes, it was possible within the existing functionality of imagefromfile, and even documented at HowTo :-). However, the documentation did not really draw attention to it. Moreover, some care needs to be taken with the margin values to ensure that things look nice. I have now added some more specific guidance to the HowTo around this, and fixed the examples using imagefromfile at AdS-QCD correspondence. There are some graphics at this page in the second half of it which still do not use imagefromfile; I think this would be good to change, as imagefromfile ensures that the caption is correctly handled, also semantically.

    Previously the caption did not allow a link within it. I have now made this possible. I also happened to notice whilst browsing the page that the nLab did not have styling for headers of depth h5; this is now fixed too in the same commit.

    • CommentRowNumber27.
    • CommentAuthorRichard Williamson
    • CommentTimeMar 14th 2020
    • (edited Mar 14th 2020)

    There are also some headers of depth h6 at AdS-QCD correspondence. These not only had no styling but were not picked up by the table of contents previously. I have now changed this, i.e. both added styling and ensured that they are picked up by the table of contents. I hope that this does not break anything on other pages; there may have been some reason why I disallowed h6 from the table of contents originally! I think possibly there were some cases of strange formatting involving h6, but hopefully these have been cleared up in the meantime. If anybody notices anything, just raise here :-).

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeMar 15th 2020

    Thanks!! Nice.

    Sure, I will change the remaining image inclusions to imagefromfile, too.

  10. Hi, thank you for h6 again! I would bring up two things:

    1. The compiler reformats when you jump over levels—I mean, when you type this

      ##

      ####

      ###

      it becomes

      ##

      ###

      ##

      I see the reason for this but it becomes inconvenient, when you want to reflect the structure of certain books like the Critique of Pure Reason.

    2. Maybe the table of contents would become more compact, if the font-size (and spacing) of level h6 would be smaller.

  11. Sorry again for my posting mess: As for troubles with h6. I don’t think that it is a feature that the “Theorem” shows up in the table of contents at HowTo#latex_syntax

    • CommentRowNumber31.
    • CommentAuthorUrs
    • CommentTimeMar 16th 2020

    Richard,

    apparently now all Definition/Theorem/Proof-environments are being picked up by the table of contents.

    A clear example is now the entry determinant (which I just edited and saved, causing the problem to appear).

    I see now that this is also what Daniel is referring to in #30 above.

    • CommentRowNumber32.
    • CommentAuthorRichard Williamson
    • CommentTimeMar 16th 2020
    • (edited Mar 16th 2020)

    Yes, that must have been the reason I did not permit h6 originally. I will fix this as soon as I get chance one way or another. But I do not have the opportunity for the moment; perhaps this evening European time.

    • CommentRowNumber33.
    • CommentAuthorRichard Williamson
    • CommentTimeMar 16th 2020
    • (edited Mar 16th 2020)

    I have fixed this now, and re-rendered manually all pages edited since the changes announced in #26-#27 were made (hopefully I didn’t miss any).

    Regarding point 1 in #29, thanks for raising Daniel, but for now I don’t think I’ll change this. I explicitly wished to disallow level jumping when I wrote the table of contents parser (for some pages there was a mess previously coming from the fact that it was not enforced), and I think probably it should still be enforced; in a book, one would never hop level for example. What you describe is, it seems to me, a corner case a) which would be solved if all intermediary sections from the book were named, and b) which does not seem disastrous to me even with the level hopping.

  12. Regarding point 2 in #29, thanks again for raising. Yes, the table of contents does sprawl in some cases, it probably would be good to try to make it a little more compact one way or another. Not sure about making h6 smaller, but it is an idea. For now I’ll not prioritise this unless others feel that it should be acted on reasonably soon, but it’ll be listed here, and we can come back to it as time allows.

    • CommentRowNumber35.
    • CommentAuthorUrs
    • CommentTimeMar 30th 2020

    Hi Richard,

    concerning again the imagefromfile-functionality:

    I would like to reduce the vertical distance between the image and its caption a little.

    I have tried reducing the bottom margin, but apparently that is counted from below the caption.

    Is it possible to adjust the caption positioning itself?

    Or maybe it’s good for it to be hard-coded, but then it could maybe be a bit smaller.

    • CommentRowNumber36.
    • CommentAuthorGuest
    • CommentTimeApr 7th 2020
    I'm not sure where am I supposed to put this but the link https://ncatlab.org/nlab/show/geometry+of+physics+--+representations+and+associated+bundles shows an XML error .
    • CommentRowNumber37.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 7th 2020
    • (edited Apr 7th 2020)

    The doctype declaration on top of the file https://ncatlab.org/nlab/show/geometry+of+physics+–+representations+and+associated+bundles says:

    <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" > <html xmlns="http://www.w3.org/1999/xhtml">

    XHTML 1.1 is an anachronism. We should probably replace it with a modern doctype:

    <!doctype html> <html>

    • CommentRowNumber38.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 7th 2020

    XHTML 1.1 was published in May 2001 (!), which reminds me how extremely old Instiki is.

    • CommentRowNumber39.
    • CommentAuthorRichard Williamson
    • CommentTimeApr 7th 2020
    • (edited Apr 7th 2020)

    Re #36, thanks for raising! There was a problem with inconsistent section depths on this page, which led to the table of contents not being able to be parsed. I have attempted to correct it; the page displays now, but please check, Urs, if the sections are now at the correct depths (i.e. that things which are sections should indeed be sections and not sub-sections, etc).

    Re #37-38: the date in the header is still OK for XHTML, Instiki is not that old :-). But yes, we can consider replacing the header. Recently I have been making progress on my long-term project of rendering Instiki itself obsolete when it comes to the nLab; much of the important code already does not belong to Instiki. Fairly soon, I hope that a completely new nForum will be ready for use with a new, fast and effiicient, Markdown parser written in C, as the first major visible sign of this latest work. This is a step towards moving the nLab/nForum to the cloud; I do not intend to move it as-is, it will be moved once the software is more amenable to it.

    Re #35: I have tweaked this now. Let me know if it is OK now.

    By the way, may I suggest to consider whether it is a good idea to use floated right images in general. E.g. the first graphic on the page AdS-QCD correspondence is not fully visible on a mobile phone, or at least my mobile phone. I don’t think one typically loses much by simply have an image be placed vertically somewhere below the parapraph which refers to it.

    • CommentRowNumber40.
    • CommentAuthorUrs
    • CommentTimeApr 8th 2020

    Thanks!

    Re #35: On my phone everything looks perfect when I hold the phone horizontally.

    (When I hold it vertically, the floating images are placed as ordinary non-floating images, the only problem now being that they get aligned to the right margin of the text (instead of lapping over it as the non’floating images do) and then partially flow out of the screen to the left if they are large.)

  13. Ah, I never use my phone horizontally! If you are fine with how it is now, then all is good :-).

    • CommentRowNumber42.
    • CommentAuthorUrs
    • CommentTimeApr 8th 2020

    But wait, it seems this issue shouldn’t depend on whether the images float or not:

    In vertical phone view (or generally when the window has not enought horizontal space to allow floating, I suppose) the floating images get placed vertically just as the non-floating images do, and it shouldn’t make a difference.

    The problem (on my phone at least) arises only because, as soon as horizontal space is lacking, the floating images lap to the left, while the non-floating lap to the right, while the phone shows no whitespace to the left of the text margin, but loads of whitespace to the right of the text margin.

    This need not be so, does it? It seems unrelated to the floating functionality.

    • CommentRowNumber43.
    • CommentAuthorUrs
    • CommentTimeApr 10th 2020
    • (edited Apr 10th 2020)

    Hi Richard,

    there is a user here having trouble with the spam filter when editing Kakeya Conjecture. Can you see what’s causing the problem?

  14. Re #43: I have replied on that thread now.

    Re #42: I have fixed this now. The fix will only apply upon re-rendering a page, but I have done that for AdS-QCD correspondence, so this page should display fine on a mobile phone held vertically now. The problem was in fact related to floating, basically the fix disables floating (and adds a small bit of padding on top) when on a mobile phone, or when the window width is sufficiently small. All that said, I still personally feel that it typically looks better not to use floating, and just to position tables vertically; but of course it is up to you, I am no expert on typography!

    • CommentRowNumber45.
    • CommentAuthorDmitri Pavlov
    • CommentTimeApr 13th 2020
    • (edited Apr 13th 2020)

    The current revisions of constructive model structure on simplicial sets has a strange rendering bug. This bug only shows on the main page linked above, but strangely not on the revision page https://ncatlab.org/nlab/revision/constructive+model+structure+on+simplicial+sets/5 (notice that 5 is the number of the newest revision).

    Specifically, in the section “Weak equivalences”, the second list item appears to have a spurious paragraph break in the middle of a list item. But this paragraph break does not appear on the revisions page.

    • CommentRowNumber46.
    • CommentAuthorRichard Williamson
    • CommentTimeApr 13th 2020
    • (edited Apr 13th 2020)

    There was some weird character in the source, a carriage return I guess. I fixed it by just making the source line continuous, as it should be. Probably such carriage returns should be tolerated and ignored/corrected, but one should definitely avoid carriage returns in this situation anyway and not rely on the server handling it, so I’ll not fix it for now. It is almost certainly an artefact of the interplay between the old and new renderers, and should be gone once we are moved off the old Instiki completely.

    Only the current page uses the new renderer, not any of the revision pages (including the current one). Again, I do not plan to change this until the old Instiki is gone.

  15. With regard to this discussion: There is XyJax which is an extension for MathJax that renders Xy-pic’s.

    • CommentRowNumber48.
    • CommentAuthorUrs
    • CommentTimeMay 7th 2020
    • (edited May 7th 2020)

    This bug must have been listed here before, but Dmitri brings it up again here:

    A hyperlink followed by a whitespace followed by another hyperlink gets rendered without the whitespace.

    (Minimal example currently in first line of Sandbox.)

    • CommentRowNumber49.
    • CommentAuthorvarkor
    • CommentTimeMay 8th 2020
    • (edited May 8th 2020)

    Formatting, such as italicised text, inside hyperlinks doesn’t seem to work. See *normal* on this page, for instance.

    • CommentRowNumber50.
    • CommentAuthorvarkor
    • CommentTimeMay 8th 2020

    Could the syntax for links to other pages be added, as suggested by this comment (that also has some other suggestions)?

    Also, it would be quite nice to support Markdown for internal references instead of using the [[ page name | text ]] style, which is entirely different to the style of internal links. I’m not sure if there is precedent for this elsewhere, but something like [text](§page name) would feel far more consistent with the existing Markdown notation. Is this something anyone else has ever wanted?

    • CommentRowNumber51.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 14th 2020

    Are there any plans to add support for \cite{…} and \bibitem{…}?

    I am currently using them (e.g., at numerable open cover) in anticipation that such support will be added.

    I thought that translating the \cite/\bibitem syntax into Maruku’s syntax is a matter of simple substitution, or is there more to it?

    • CommentRowNumber52.
    • CommentAuthorvarkor
    • CommentTimeMay 15th 2020

    Is it possible to redirect to a specific section of a page? E.g. “higher-order abstract syntax” should be redirected to this section, but at the moment, it’s just redirected to the page in general.

    • CommentRowNumber53.
    • CommentAuthorUrs
    • CommentTimeMay 15th 2020
    • (edited May 15th 2020)

    No, that’s not possible. But to get a hyperlink to a specific section in a page, give it an anchor name by

      ## A section
        {#ASection}
    

    and then refer to it by

      see [here](PageName#ASection)
    
    • CommentRowNumber54.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 15th 2020

    Is it possible to rename nForum posts?

    For instance, this post https://nforum.ncatlab.org/discussion/3179/measurable-locales/#Item_7 is title “Measurable locales”, which means that when somebody edits the article measurable locale, the software will create a new thread, ignoring the old one.

    • CommentRowNumber55.
    • CommentAuthorUrs
    • CommentTimeMay 15th 2020

    Yes, but only by the author of the first message in the thread:

    Clicking “edit” on that first message (which it’s original author can do) gives an edit pane for the message itself, but also for the title of the thread.

  16. Just a quick note that I have seen the above posts, thanks for raising. I have been tied up with other things and not been able to take a look yet.

    • CommentRowNumber57.
    • CommentAuthorMike Shulman
    • CommentTimeMay 19th 2020

    The page cartesian bicategory appears to be uneditable: upon save I get “Invalid LaTeX Block:” with no further information.

    • CommentRowNumber58.
    • CommentAuthorRichard Williamson
    • CommentTimeMay 20th 2020
    • (edited May 20th 2020)

    This usually occurs when LaTeX is not closed properly, i.e. missing a dollar sign. The old Instiki permitted this (with consequent erroneous rendering of the page), the ’new’/current one does not. I will take a look when I get a chance.

  17. Fixed now. In this case the problem was that someone had tried to include vertical space by an empty LaTeX block, which I removed and replaced by \linebreak.

    • CommentRowNumber60.
    • CommentAuthorRichard Williamson
    • CommentTimeMay 20th 2020
    • (edited May 20th 2020)

    Re #54 and #55, I have fixed this in the database now. I do this whenever I see this problem occurring, merging the newly created thread with the old one. Another case where it occurs is where the original thread has a different category than ’Latest changes’.

    Just post here if you see other examples of this which I overlook.

  18. Re #48, this is a bug in Maruku, the markdown parser which Instiki uses. I will see if I can find a way around it when I get a chance.

    • CommentRowNumber62.
    • CommentAuthorMike Shulman
    • CommentTimeMay 20th 2020

    Re #59 Thanks!!

    • CommentRowNumber63.
    • CommentAuthorUrs
    • CommentTimeMay 26th 2020
    • (edited May 26th 2020)

    Hi Richard,

    does the imagefromfile-environment admit specifying a URL to make the image be hyperlinked?

    (I have tried some evident choices for tags, such as URL and href, without success)

    • CommentRowNumber64.
    • CommentAuthorUrs
    • CommentTimeMay 26th 2020

    Hi Richard,

    on a related note, I just encountered a bug with the imagefromfile-environment:

    A minimal example is now in the Sandbox:

    The source code of the Sandbox page right now is this:

      \begin{center}
      \begin{imagefromfile}
          "file_name": "HardWallModelPredictions.jpg",
          "width": 440,
          "caption": "From [Erlich 09, section 1.2](#Erlich09)"
      \end{imagefromfile}
      \end{center}
    

    while the output of the Sandbox page is this error message:

      XML Parsing Error: mismatched tag. Expected: </div>.
      Location: https://ncatlab.org/nlab/show/Sandbox
      Line Number 314, Column 35:
      <div style="text-align: center"></p>
      ----------------------------------^
    

    The error goes away when removing either (or both, of course) of

    1. the center-environment,

    2. the image caption.

    • CommentRowNumber65.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2020

    Hi Richard,

    I have now a “working” example of the bug with modifying the redirects:

    I have just started to add a new subsection to diffeological space. In the source code it’s visible as

      ### Relation to topological spaces
    

    Then I added one redirect to the page

      [[!redirects D-topology]]
    

    After that, the added material no longer appears in the output of the page.

    \,

    Experience shows that if next I made any other edit to the page the output will appear. But I’ll leave it as is for the moment in order to demonstrate the bug.

  19. Thank you! I’ll take a look at these things as soon as I get a chance.

    • CommentRowNumber67.
    • CommentAuthorvarkor
    • CommentTimeJun 3rd 2020
    • (edited Jun 3rd 2020)

    Recently, when visiting the nLab from Google, extra text has been added to the URL, and the corresponding text highlighted on the nLab, which seems to slow everything down, and also looks ugly. An example of what I mean:

    https://ncatlab.org/nlab/show/over+category#:~:text=Definition,-The%20slice%20category&text=The%20slice%20category%20is%20a,g%3AX%E2%86%92X%E2%80%B2

    I’m not sure who’s responsible (Google/nLab). Is anyone else experiencing this?

  20. Hi, I’ve not noticed this myself. There have not been any changes to the nLab software which should lead to this. I also could not reproduce it in the browsers I use. Let me know if it looks like there is something we can do on the nLab side about it.

    • CommentRowNumber69.
    • CommentAuthorvarkor
    • CommentTimeJun 3rd 2020

    It’s been happening sporadically and I could definitely believe it was Google A/B testing something on me that wasn’t working as intended.

  21. Footnotes increase line height, e.g. see Giry monad. Reason is that line height in normal text is not 1em but 1.3em. Thus the <sup>-tag shifts this extra space to the top creating too much space. Remedy: sup { line-height: 1em; }.

  22. Thanks for notifying! I made the change you suggested; it looks better at Giry monad now, but I guess not perfect yet. I ran out of time to experiment further; will do so when I get the chance.

  23. I find spacing at the beginning of lists in lists not pleasing, e.g. lax functor. I would suggest li li:first-child { margin-top:.25em; }.

  24. I have to correct my suggestion:

     li ul, li ol { margin-top:.25em; }
     li li: { margin-bottom:.25em; }
    

    The is another point about first order lists. There is too much spacing when a list follows a paragraph, e.g.pseudofunctor. My suggestion

    p { margin-bottom:.5em; }
    ul, ol { margin-top:.5em; }
    
    • CommentRowNumber74.
    • CommentAuthorDaniel Luckhardt
    • CommentTimeJun 23rd 2020
    • (edited Jun 23rd 2020)

    Speaking of margins, I would also suggest

    h6 { 
        margin-top:2em; 
        margin-bottom:1em; 
    }
    

    The only two examples using h6 I am aware of are Science of Logic and Critique of Pure Reason.

    • CommentRowNumber75.
    • CommentAuthorDaniel Luckhardt
    • CommentTimeJun 23rd 2020
    • (edited Jun 24th 2020)

    Sorry, I correct all my suggestions from #73 again to:

      p { margin-bottom:.5em; }
      li p { margin-top:.5em; }
    

    and deleting

      li p { margin-bottom:.5em; }
    

    The problem with the current margin comes from the fact that the text in lists is put in ps (in contrast to how tables of content are generated). But ps have top/bottom margins =0.

    • CommentRowNumber76.
    • CommentAuthorGuest
    • CommentTimeJul 1st 2020

    I can’t edit because of “invalid LaTeX block” in space and quantity. I saved the edit I was trying to make as follows. Also it would be really good if there’s a preview button in the editors.

    “‘ which underlies much of mathematics is at its heart controlled by the following elementary category theoretic reasoning:

    Let SS be some category whose objects we want to think of as certain simple spaces on which we want to model more general kinds of spaces. For instance S=ΔS = \Delta, the simplicial category, or S=S = CartSp, the category of n\mathbb{R}^n and smooth maps between them.

    An ordinary manifold, for instance, is a space required to be locally isomorphic to an object in S=CartSpS = CartSp. But more generally, a space XX modeled on SS need only be probeable by objects of SS, giving a rule with which, to each test object USU \in S, we assign the set of probing maps from UU to XX, such that this assignment is well-behaved with respect to morphisms in SS. Such an assignment is nothing but a presheaf on SS, i.e. a contravariant functor

    X:S opSet. X : S^{op} \to Set \,.

    Therefore general spaces modeled on SS are nothing but presheaves on SS:

    Spaces S:=PSh(S). Spaces_S := PSh(S) \,.

    Of course this is an extremely general notion of spaces modeled on SS.

    For example, any smooth manifold MM is a presheaf on CartSp by F M:=Hom Man(,M):CartSp opSetF_M:= Hom_{Man}(-, M): CartSp^{op} \leftarrow Set, where we consider CartSpasafullsubcategoryofMan,thecategoryof[[smoothmanifold|smoothmanifolds]]andsmoothmapsbetweenthem. as a full subcategory of Man, the category of [[smooth manifold|smooth manifolds]] and smooth maps between them. F_Mmapseach maps each \mathbb{R}^ntothesetofallthesmoothwaysthat to the set of all the smooth ways that \mathbb{R}^ncanprobe can probe M$.

    In particular, every object in SS is a space modeled on SS, by the Yoneda embedding SSpaces SS \hookrightarrow Spaces_S, whereby every object XX in SS is embedded as Hom S(,X)Hom_S(-, X). That is, any object in SS is nothing but a consistent way to be probed by all the objects in SS.

    Now take a space XX modeled on SS, and consider the set of quantities on XX with values in USU \in S. It should be

    C(X,U):=Hom Spaces S(X,U). C(X,U) := Hom_{Spaces_S}(X,U) \,.

    This defines a covariant functor C(X):=Hom Spaces S(X,):SSetsC(X) := Hom_{Spaces_S}(X,-): S \to Sets. More generally, we can consider the S-valued quantities on XX to be a copresheaf on SS, namely a covariant functor

    C(X):SSets. C(X): S \to Sets \,.

    One can think of C(X)C(X) as a generalized quantity which may be co-probed by objects of SS.

    In this vein, one can say, generally, that copresheaves on SS are generalized quantities modeled on SS, and we write

    Quantities S:=CoPSh(S). Quantities_S := CoPSh(S) \,.

    Given any such generalized quantity AQuantities SA \in Quantities_S, we can ask which generalized space it behaves like the algebra of functions on. This generalized space should be called Spec(A)Spec(A) and can be defined as a presheaf by the assignment

    Spec(A):UHom Quantities S(A,C(U)). Spec(A) : U \mapsto Hom_{Quantities_S}(A, C(U)) \,.

    In total this yields an adjoint pair of contravariant functors between generalized spaces and generalized quantities:

    Spaces S opSpec()C()Quantities S. Spaces_S^{op} \stackrel{\stackrel{C(-)}{\to}}{\stackrel{Spec(-)}{\leftarrow}} Quantities_S \,.

    (That this is an adjunction can be understood as a special case of abstract Stone duality induced by a dualizing object.)

    Lawvere refers to this adjoint pair as Isbell conjugation.

    “‘

    • CommentRowNumber77.
    • CommentAuthorGuest
    • CommentTimeJul 1st 2020
    Ah, now I find the problem.

    -- Yuxi Liu
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