nLab 2009 May changes

Archive of changes made during May 2009. The substantive content of this page should not be altered. For past versions of this page beyond its own history, start here and work backwards.






  • Bruce filled in a small thing at semisimple category.

  • Urs:

    • created path integral

    • added to sieve a – potentially long-winded – lemma with detailed proof that spells out details of the fact that every subfunctor is the coimage of a morphism out of a coproduct of representables

    • created separated presheaf

    • further worked on the section “in terms of geometric embedding” at sheaf


  • Toby Bartels: Copied comments on Google and page names to the Café here.

  • Urs

    • expanded and restructured sieve – introduced the explicit distinction between the notion of sieve and that of subfunctor and made their bijection explicit: this seems to be the standard way that “sieve” is used in the literature: authors don’t seem to say “sieve” synonymously with “subfunctor”, but say “sieve” if they explicitly mean the set of elements of the subfunctor. There is a bijection, but not an equality of concepts.

    • further expanded image and coimage

    • edited and replied at semisimple category

  • Bruce responded to comments at semisimple category…and then got stuck. Can anyone help? Update from Future Bruce to Past Bruce: solved.

  • Zoran Skoda: as far as google (and other search machine’s) counterarguments of Toby (see down 22. may) I still disagree. I do care if our stuff is well indexed and hence better used by anybody looking for answer at the google including us; and I do not care about vanity issues of pro or contra google movements. If somebody gets directed to a less relevant page this is creating noise, and showing a less convenient side of our work. If effectivity is not important why are we doing this ?

  • Urs

    • further developed the section “In terms of geometric embedding” at sheaf, but will continue later. I am lacking (at least) one argument in one of the proofs currently…

    • created Bertrand Toën (and Bertrand Toen as a redirect)



  • Toby Bartels:
    • Created Banach spaces. I need to at least describe products and coproducts in BanBan.

    • Another example, Banach spaces, at internal hom.

    • Looking at other things that Urs has done, the problem may be this: If you want to put an equation within a bullet list, then you either have to put no blank lines between the equation and the surrounding text

      likethis like this

      or put enough spacing before the equation to match the indentation of the list

      likethis like this

      or, of course, both

      likethis like this

      but it will fall outside of the bullet list if you do neither

likethis like this
  • Bruce:
    • Added example of the internal hom in super vector spaces to internal hom.
    • Urs: I tried to understand what went wrong with your query box example for Tychonoff theorem but I think the roll-back has removed the original problem from the records.


  • Toby Bartels:

    • Expanded centipede mathematics and negative thinking, which David Corfield created yesterday.
    • In response to Zoran immediately below, I don't think that we should change what works for us to fit Google. It's Google's job to give good search results, not ours to optimise Google's finding us, which Google tries to prevent anyway. That said, I agree with you about the page names, for other reasons. (Note to all: There is discussion here.)
  • Zoran Škoda: created comorphism (in sheaf theory); there is hard to find a sensible and comprehensive account in the literature. Note that the treatment is more general than in the usage for the case of ringed spaces. By the way, the server is very erratic tonight, and having sometimes responses delayed by 5-10 minutes. On the other hand I strongly disagree with the changes of the names of entries massively being done today by Eric: he moves infinity-category into \infty-category. Though this is graphically appealing, google and others put higher in search results items which have search name in the title, and the entries they index are index basically by the ascii. I want to see nlab entries high in the google search, this makes our effort more useful. Fancy graphics can be WITHIN the entry, and prefereably in this decade still not in the title. I would do the redirects in the symbol variant of the title instead!! What the others think ?

  • Urs

    • I tried to fix the query box layout at Tychonoff theorem only to find that there is apparently some software clash between the query box syntax and some symbols used. So I “rolled back”. Hopefully the entry is now again in the form that Todd and Body left it. Please check.

    • started filling something into n-categorical physics just so the page looks less blank – but didn’t really have the energy or intention to produce anything of more than vaguely suggestive nature so far

  • Zoran Škoda made changes to pure motives. Please do not use defined term algebraic spaces when it is not appropriate.

  • David added exposition references to pure motive

  • Urs

    • added explicit description of colimits in SetSet to limits and colimits by example – I have a dumb question: frequently I want to use displayed math equations in a bullet list item. But more often than not when I try this the parser gets very mixed up and produces weirly formatted output – what am I doing wrong??


  • Bruce added link to Evan Jenkins’ notes of the Northwestern TFT Conference 2009, and changed text+math font on the nLab to a serif theme by editing the CSS. This will only be for one day, just to create awareness of what’s possible. Discussion on the nForum.








  • Urs:

    • added some bits to subobject classifier: a short intro, the realization in presheaf topoi and a paragraph on nn-catgorical generalizations and the interpretation of truetrue as a 1-1-category.

    • created power object, which is mentioned in the alternative definition at topos – guessed the definition from the Axiom of power sets at Trimble on ETCS I, but I am not really sure – somebody please have a look and check

    • expanded and restructured Grothendieck topos

    • added what should be a detailed proof to geometric embedding that for every geometric embedding f *:FEf_* : F \hookrightarrow E the category FF is equivalent to the localization of EE at the morphisms that are sent by f *f^* to isos

      • there are two possibilities here: either I am mixed up or this is somewhere in the literature. If the latter, can anyone give me a pointer to a reference that mentions this explicitly?
    • created geometric embedding – am in the process of filling in a detailed discussion of the relation to localization

    • added a paragraph to localization describing and emphasizing the point that lots of localizations one runs into in practice are reflective subcategories or actually geometric embeddings. In particular Bousfield localization presents the corresponding reflective (infinity,1)-subcategory. I am still not really satisfied with the entry localization, though, I am hoping we can eventually present the conceptual basis here more clearly. After all, the perspective on sheaves and sheafification in terms of localization/geometric embedding has been shown to be the workable road to \infty-stacks, which, as tradition has it, are well worth pursuing in general and on an nnLab in particular :-) It is curious that there seems to be a cultural divide in the literature here: the book by Kashiwara-Schapira for instance amplifies sheafification as localization, which paves the road for \infty-stacks presented by the model structure on simplicial presheaves, while the book by MacLane-Moerdijk amplifies sheafification as geometric embedding, which paves the road for the simple definition of (infinity,1)-sheafification by reflective (infinity,1)-subcategory. Lurie’s book effectively gives the unified perspective, which I think is worthwhile presenting very clearly here on the nnLab, since it is coceptually so simple and transparent and in practice so powerful. Over.

    • added example to localization of a simplicial model category?

    • added a little proposition to dependent product and a little example to geometric morphism

  • Tim: I have given a partial reply to Urs question at homotopy coherent nerve.




  • Toby Bartels: Fixed the last ‘heuristic’, at local system, among other such fixes. (You gotta open them up for editing and look at the source!)

  • Urs:

    • tied up the loose end at descent using the expected result that Dominic Verity was kind enough to proof on request and then confirm by private communication
  • Andrew: (I forget whether I’m supposed to add this to my earlier comment, or here, or add a word of this to each of the other entries of the day). I’ve created SVG Sandbox? expressly for the purpose of mucking about with SVGs to get them to look right. My rationale is explained at the top of that page, together with some suggestions on how it might work. The point is that one SVG can be rather large and I think that putting them in the regular Sandbox to test stuff is a bit anti-social. Hopefully we can get stuff in the FAQ and HowTo on good ways of importing SVGs as there seem to be a few “special features”. I’ve shifted the recent SVG-related stuff from the original Sandbox to the SVG Sandbox?, but I shifted the discussion that Bruce started to the nForum.

    • Urs: I believe Mike had requested that we announce each change here always on the very top of the list, even if we had earlier logs the same day with other people’s logs already on top of them. Seems to be a reasonable practice if the point of this page here is to alert others of changes. Which it is. So, yes, the way you did it should be the preferred way.
  • Urs:

  • Bruce: Added discussion to Sandbox about Andrew’s TikZ->SVG method. Maybe we can get good and easy graphics going soon in Instiki!

  • Urs:

  • Andrew: Finished off the heuristic shift. Bizarrely, one page, local system, comes up in the Instiki search as having “heuristic” in it but I can’t find it. The other instances of “heuristic” that are left are either correct or within discussions.

  • Urs:

  • Toby Bartels:





  • Urs

    • worked on end: added a section “End as equalizer” where I try to motivate the formula for the end over VV-valued functors from the equalizer formula for limits, then give the equalizer formula for ends – personally I find this a bit more helpful than dinatural transformations, but that is certainly just my ignorance – I added an “Idea” and a “References” and an “Examples” section

    • added the definition to enriched category (yes, that was still “left as an exercise”)




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Last revised on August 24, 2015 at 02:30:12. See the history of this page for a list of all contributions to it.