nLab 2009 July changes

Archive of changes made during July 2009. The substantive content of this page should not be altered.





  • John Baez: warning: I believe “special lambda-ring” is an old-fashioned name for what almost everyone now calls a lambda-ring. This is explained by Hazewinkel in his article cited on lambda-ring. So, I do not believe we should have a separate article on special lambda-ring.

  • Zoran: created Dennis trace (experts please expand!)

  • Toby Bartels:

    • Incorporated results of discussion into effective epimorphism and regular epimorphism.

    • Everyone who uses ‘\sqcup’ (\sqcup) should be aware of ‘⨿\amalg’ (\amalg) and ‘\coprod’ (\coprod), as in ‘A⨿BA \amalg B’ and ‘ iA i\coprod_i A_i’.

    • Interaction with Rafael Borowiecki at category theory.

    • Golf department? Golf department??

      • Urs Schreiber: aha, I didn’t get that either – but figured it must be me missing some English thing
  • Urs Schreiber created Kan fibrant replacement

  • Zoran: created Grothendieck Festschrift, and quoted it as an addition to the timeline entry.

  • Urs Schreiber

  • Lab Elf (service department): The problem was that there was a reference to a theorem in the section that you wanted to remove. By removing that section, you were trying to create a non-existent link. By reformatting that sentence, I was able to remove the section. Another Lab Elf (golf department) may be along later to clean up the links since I just did what was necessary to remove the section.

  • Urs Schreiber

    • split off Reedy model structure from Reedy category – added an “Idea” section and started expanding on some of the technical details

      by the way: it’s funny I can’t remove the old section “Model structures” from Reedy category which is now reproduced and expanded at Reedy model structure: I always get an “Internal server error” when I try to do that. I am familiar with the occurence of this error when one adds certain things (such as double-dollar included displyed math without line breaks before and after) but I couldn’t figure out which problem the removal of that paragraph causes. So that section is still sitting there, duplicated now.

    • added clarifying remarks to the references at K-theory

    • in a similar vein to the below comment i added to the beginning of delooping a remark how the one-object groupoid BG\mathbf{B}G and the classifying space BGB G are the same object under the homotopy hypothesis

    • from private discussion with somebody it became clear to me that the entry homotopy hypothesis failed to get across one of the main points with the required emphasis. I now added the central theorem about the Quillen equivalence between Top and SSet right at the beginning. The disucssion of all the subtleties and generalizations should come after that.

  • Andrew Stacey: Mathematical and non-mathematical stuff going on at Tall-Wraith monoids. Folded up the mathematical bit of the middle discussion (on what’s so special about AbGrpAbGrp) into the main text, but probably not in the nicest and clearest way. Also continued the discussion on fonts and the like further down.

  • Urs Schreiber

    • added a few words to (infinity,1)-topos (in particular pointed out that the equivalence of the two definitions given is a main theorem, and added the links to the entries on models)



  • Toby Bartels: Lots of changes (mostly additions) to measure space. Please see if the notation is comprehensible. I have to check on a couple of things, but I left query boxes. There are several variations, but I only included things that people can actually get tenure by studying. No centipede mathematics just for the sake of it! (well, except for one comment, appropriately linked).

  • Andrew Stacey: continued the sparring at Tall-Wraith monoid (and answered the serious query). I wish I’d known the fascination with centipedes earlier, we caught one today and I could have gotten a good picture of it.

  • John Baez: inserted centipedes in quasigroup, magma, and the section on weakened definitions in group. Made a few other small changes in these.

  • Toby Bartels:

  • John Baez: meddled a bit with centipede mathematics. Accidentally created a page called ‘semigroups’ — sorry, Toby; it looks like you’re merging it with semigroup, which said a lot of the same stuff in a more sophisticated lingo.

  • Mike: A couple of comments at small object argument.

  • Toby Bartels:

  • John Baez

    • asked a question about notation near the top of Tall-Wraith monoid, and tried to polish the proof that a Tall-Wraith monoid in abelian groups is just a ring, and enjoyed bickering a bit more with Andrew Stacey in the big green box near the bottom.

    • had some fun with centipede mathematics - see also my reply to Toby below.

    • deleted query by Rafael over on category, which had been answered by me and untouched for a while. He’d asked about ‘categories as 1d CW complexes’, but I think the item on categories as ‘directed graphs with composition law’ now answers that — even for people who don’t know what a CW complex is.

    • reduced the number of appearances of the word ‘isic’ over on isomorphisms; while it’s fun to make up new jargon, I don’t think we should actually use ‘isic’ when explaining concepts when ‘invertible’ will do. We don’t want to convey an impression of quirkiness, and we don’t want to require the reader to look through the whole page to understand new jargon when well-known jargon already exists.

    • deleted discussion by Toby and Tom over at regular monomorphism, since Tom said it was okay to do so, and some time has passed.

  • Todd Trimble wrote generalized multicategory, and added a reference at Crans-Gray tensor product to Sjoerd Crans’s papers. Guessed that his papers on teisi might be relevant to an inquiry Mike Shulman made there.

  • Toby Bartels wrote separation axioms (and a stub at disjoint sets).


  • John Baez wrote comments on isomorphism, bicategory, measure space, and under a July 24 comment of Eric's here. Toby has responded to all of them except the one at bicategory.

  • Toby Bartels:

    • As long as we're having a conversation, it should be safe for Eric to assume that I've responded to him if I post something here, even if I don't mention him.
    • A brief list of examples, mostly for the purposes of terminology, at isomorphism.
  • Eric: Responded to Toby at measure space and Densitized Pseudo Twisted Forms.

  • Tim: I have started an entry on HQFTs. Initially this will summarise Turaev’s theory, but I hope to get a bit more daring later on. I hope someone will tell me (then) if I am talking through my hat. (I rarely wear one.)

  • Toby Bartels: Comments for Eric at measure space and on his web.


  • Toby Bartels:

  • Eric:

  • Urs Schreiber

  • Eric:

    • Added more to questions on measure space. Whenever I see a long convoluted definition, e.g. measurable space, I tend to think there should be some short, concise, arrow theoretic description that incorporates all the little factoids into one pretty picture. A wild guess (that I know is wrong, but hopefully inspires someone to write down what is right): a measurable space is some kind of presheaf or maybe a representation on ????.

    • John Baez: the definition of measurable space is pretty darn simple and quick: it’s a set with a collection of subsets that’s closed under complements and countable unions. Such a collection is usually called a σ\sigma-algebra, and all this is explained pretty early on in the page measurable space. Whoever wrote the longer discussion below was just having fun analyzing the definition into little bite-sized pieces (I have my guess as to who this might be.)

      • Toby Bartels: Guilty as charged.

      • John Baez: The guilty conscience need not be accused by name. I think we should warn the reader when we go off on an excursion like this. Perhaps just a warning like: The following passage might be considered centipede mathematics, together with a small version of the following picture. I wish I knew how to center a picture!

        A pic

  • Tom Ellis? created extremal monomorphism

  • Urs Schreiber

    • edited algebraic K-theory a bit

    • created K-theory – as opposed to my previous take on this which was then moved to the “Idea” section at topological K-theory this time this is aiming for the fully general bird’s eye picture with indications how that produces all the special realizations in special cases

    • created decategorification – evidently much more can be said here, but it’s a start

    • expanded the “Idea” section at spectrum and effectively rewrote it – added a link to combinatorial spectrum at the end, which probably should be thought of as a concrete realization of the idea of \mathbb{Z}-category – accordingly I changed the title of the last section from “Conjectures” to “Combinatorial models”.

    • added a reference to Weibel’s online book to algebraic K-theory

  • Andrew Stacey: Tried answering John’s questions over at Tall-Wraith monoid. Probably lots for the lab elves? to work on there.

    (I confess that I did have the Hogwartian house elves uppermost in mind, but the shoemaker elves were not far off either. Being now in the Nordic realm I probably should have said ‘lab troll’ but trolls already have a place on the internet and it is Not Here)

  • Urs Schreiber

    • started putting material into brane – as a preparation for something I plan at K-theory
  • John Baez: I answered some remarks by Mike Stay and Eric over on free cocompletion. I also had an hour-long chat with Mike that should eventually push this exposition forward quite a bit: I explained coends to him, which is a lot easier in words than on paper. But I hope we get the explanation into the nnLab eventually!

  • Toby Bartels: Answered Eric's first question; I'm not ready to think about the second one yet.

  • Eric: Asked some questions on measure space.


  • Toby Bartels: Remarks on notation at measure space.

  • Ben Webster created Hecke algebra

  • Zoran ?koda?: created Dunkl operator, double derivation; it is a start of a series which should include entries on Cherednik algebras, Knizhnik-Zamolodchikov connection, Calogero-Moser system, Gauss-Manin connection, Calogero-Moser space, deformed preprojective algebras and so on…

  • Urs Schreiber:

    • effectively rewrote infinity-stack – expanding it considerably, adding the relevant pointers to all the new material that has come together since I first wrote this

      • removed the discussion of costacks entirely. I’ll turn that into a separate entry in its own right eventually

      • John’s question there had been about my notation >\to\gt – that was a hack for the symbol for a fibration, an arrow with a double tip. In my new version this no longer appear, though it may still appear at hypercover, which is linked to, and elsewhere.

        Toby knows how to typeset such arrows correcty. Maybe he could add a section to HowTo with the relevant information and links to special symbol lists

    • added a reference at (infinity,1)-functor to (infinity,1)-category of (infinity,1)-functors – the discussion of this issue that I like most is currently at models for infinity-stack (infinity,1)-toposes. Eventually that should be discussed better at the relevant entries.

  • Toby Bartels:

    • Conversation with JCMcKeown at (n,k)-transformation.
    • Created lab elf?, just a bit of fun; credit Eric Forgy (I think) for the idea.
      • Eric: Not me. I think Andrew gets credit for that. I added a link to lab elves? from How to get started :)
      • Toby: Ah yes, with his British references to Hogwarts. But the term made me think of den Brüdern Grimm instead; in fact, after reading their tale (an English version of which is linked from the new page) again, I see that Rowling's house elves owe more to it that I'd remembered!
    • Added my touch to How to get started; among other things, I didn't like the horizontal rules around the images, but maybe they were there for a reason?
  • John Baez:

  • Toby Bartels:

    • I would like to make a plea for adding entries to the top of this list. There are some comments and entries that I see only because I check every single change in the history; while obsessive behaviour may be correlated with mathematical ability, you can't expect people to see things unless you add them to the top. If that means that you give yourself two entries in one day or even report one day's events in the next day's space, then so be it! (For conversations, I suppose that it depends on whether you think that the person that you're having a conversation with will see your comment; but be aware that they might not.)
    • I wouldn't be very quick to get ban query boxes at How to get started; people need to be able to ask questions about how to get started! That doesn't really apply to discussion by the regulars about how to explain, of course, but I would want Jim Stasheff, for example, to ask questions there if he wanted to, without people trying to clean up too quickly.
  • Urs Schreiber:

    • created Dominic Verity

    • created Verity on descent for strict omega-groupoid valued presheaves

    • reworked the How to get started according to my opinions and our disucssions here

      there are now two sections, one on how to paste source code of a comment one is about to submit to the blog, the other about how to paste non-source code material

      in the course of this I have removed lots of the previous discussion on these points – the goal is to keep that particular page clean of auxiliary discussion and as brief and to the point as possiblle, because that’s the point of this page

      if anyone feels I removed too much, please use Rollback to grab the deleted material and then cancel the rollback and insert the missing material in a suitable section at the main HowTo entry

  • Andrew Stacey: I’ve banned ‘Recently Revised’ for the time being. My method of banning has probably blocked it for all the private webs as well. If that’s really annoying then let me know and I’ll try to find a more specific method of banning just the nlab one.

  • Andrew Stacey was pleasantly pleased to stumble across Tall-Wraith monoids and made a few minor alterations (mainly style, and added a couple of references). I’ll shove this question over on the forum as well, but should we have a lab convention on fonts for categories, functors, objects, and the like?

    • John Baez: especially given the large and opinionated group of contributors, we probably shouldn’t fuss over fonts too much, but I’m in favor of the KISS philosophy: “keep it simple, sweetheart”. Namely: use capital letters for big things, small letters for little things, and Greek letters when you run out of ordinary letters, or want to show off your erudition.
  • David Corfield

  • Urs Schreiber

    • renamed the new easy-basic-HowTo page to How to get started

      then I reworked the formatting and edited pieces here and there

      to Bruce Bartlett: I think on that particular page we don’t want query boxes, as that page is supposed to provide quick unambiguous information that tries to deconfuse people instead of to confuse them – please see my reply and check if you can work something into the paragraph right before the query box that allows to remove that query box

      Andrew Stacey I concur, but couldn’t delete the query box as I made a remark in it and so if I delete the box now then that would permanently remove that remark. Someone else could do it (or I could in half an hour’s time).

  • David Corfield: Started Lambda-ring with some Baezian exposition and an abstract of James Borger. Hmm, is there a difference between λ\lambda-ring and Λ\Lambda-ring? This paper uses both.

  • Toby Bartels welcomed Sebastian Thomas? at (n,k)-transformation.


  • Tim Silverman?: Answered a request from the n-Cafe by creating How to Copy and Paste Material from the n-Cafe and Include Links Back and Forth?

  • Urs Schreiber

    • slightly polsihed further at strong monad and removed the tentative-alert, now that Todd also approved of the statement

    • filled in a bit of text and some references at conformal field theory

    • added the notion of Frobenius lax-and-oplax functors to lax functor and provided pointers to their use in CFT

    • added a remark by Todd Trimble to associahedron on their relation to orientals that I asked him about by private email

  • John Baez:

    • Added more information to tensorial strength. Some of this should be checked.

    • Added more examples to lax functor. I’m in a lax mood these days, and I really enjoyed it when Paul-André Melliès told me a definition of ‘enriched category’ in terms of lax functors. This works for categories enriched over a bicategory, not just a monoidal category. Do we have any entry on enrichment over bicategories? If so, maybe someone could add a link.

      Urs Schreiber: we had some old discussion on the blog on this description of enriched categories – I used to be interested in that in the context of A Note on RCFT and Quiver Reps – I’ll maybe add something about this to the entry

  • Urs Schreiber:

    • created strong monad

    • created lax functor

      • added to monad the statement that a monad in BB is a lax functor *B{*} \to B
    • replied at (n,k)-transformation – I think that in principle this gives all the required information, but I am aware that eventually someone should describe that all explicitly in detail at that entry

  • Toby Bartels: Copied to (n,k)-transformation a question that was sent to me by email, and partially answered it. (Urs could probably answer the rest.)

  • Urs Schreiber:

  • Toby Bartels:

  • Urs Schreiber:

    • added links and references to tricategory

    • worked on the Idea section at category theory: I reformatted a bit the existing material, included lots of hyperlinks and filled in various further bits, such as a paragraph that lists the fundamental classes of examples and the quote from Barry Mitchell that Todd just mentioned on the blog

      • I was surprised to find the entry in a much more developed and pleasant state than I remembered it – maybe I missed the announcement here, or could it be that there was a major edit to the entry that wasn’t logged here at Latest Changes? Please remember to alert us here.

      • I am now hopeful that eventually we’ll be able to turn what should be the pivotal nnLab entry into something decent, too: that on higher category theory. At the moment that one is not a good advertisement of the nnLab project.

    • replied and reacted at locally presentable category

  • Eric: Asked, “What is a ‘component of a cocone’?” on

An Exercise in Kantization.

  • Urs Schreiber where did you see that term used? Maybe the question (or its answer) belongs at colimit. Do you have an idea what a cocone itself is? It consists of lots of morphisms from the objects of a diagram to the cocone tip. If we regard the cocone as a natural transformation to a constant functor, then the components of that natural transformation are these single morphism from objects to the tip of the cocone. These I would call “components of the cocone”.

  • Zoran ?koda?: Thank you Toby, your new clarifications in essential image and replete subcategory are pretty helpful and clear, and I agree with them. Still I would like to think of more clean scheme of thinking of various kinds of images internally, in connection to various kinds of factorization systems and even multistep factorizations like Postnikov systems. There is probably a framework where, despite the differences all the kinds of images including homotopy image belong. The crucial is choice of a sort of factorization system: a variety of an image is basically the second morphism in the factorization (or less precisely its domain). In higher categories sometimes multistep factorizations systems are interesting, like Postnikov towers in topology. This way it may satisfy the point of view of Urs, who was IMHO not precise at the beginning but eventually pointed in the right general direction, and the reference of Barwick which he found seems to be really useful.

  • Toby Bartels:


  • Urs Schreiber

    • after discussion with Zoran Skoda I split off homotopy image from essential image, reserving the latter for the essential image of a functor of categories – I haven’t touched the content of essential image otherwise

    • updated link list at Higher Topos Theory (mostly under Appendix/Category Theory). For what that’s worth, the appendix is now getting pretty close to being fully indexed.

    • added reference to Richard Garner’s Understanding the small object argument to small object argument

    • created transfinite composition

    • made at small object argument the theorem a formal theorem (with theorem environment and all), added a list of references and – in the paragraph that is now right before the theorem – tweaked the former assumptions a bit, which I guess were taken by Mike Shulman from Hovey’s book. My impression is that in the “modern” literature the ambient category is assumed to be locally presentable – but it would be great if an expert checked my modifications (see also the further literature that I list)

  • Andrew Stacey took the hint and started incorporating the discussion into the main thread at paracompact space.

  • Urs Schreiber

  • Zoran ?koda?: I actually do not think that Toby’s correction to essential image is correct. I mean that essential image is removing evil from image. No, image SUBcategory is just a specific and unique internal (subcategory in narrow sense) CHOICE of the (external) image of the functor within Cat as a category. Essential image subcategory is just a specific and unique choice of the bicategorical image of the functor considered as a 2-functor within Cat as a bicategory. The same with higher version. The homotopy image which Urs looks is just about image in external sense and not about the internal choice of which subset of k-cells for every k is chosen. Making a replete choice of subcategory is like taking a maximal atlas of a manifold to remove nonuniquness in the class of all atlases - so in a sense it is a maximal choice with respect to the target; the usual image of a functor is more calculated with respect to the domain of the functor. In bicategory Cat the two are equivalent; in category Cat they are not isomorphic.

    • Urs Schreiber: I am not sure I know what you mean by external vs internal. But I supppose one point you are making is that an essential image is/should be defined only up to the relevant notion of equivalence. Do you mean by “external” a characterization of essential image by a universal property, whereas by “internal” you mean a concrete representative of that (unique only up to equivalence)?

      Do we agree on what the “external” definition should be? Is it the one I suggested it should be? In that case we might reorganize the entry by startiing it with the abstract nonsense definition and then taking the replete version as one concrete realization in Cat.

    • Zoran Skoda No (I have the feeling that you are not reading what I wrote), we disagree on what external definition should be because the essential image is not a notion which is external. It is a CHOICE of literally a subcategory, not a choice of embedding of categories in abstract sense, it is a choice of a SUBSET of n-cells which is a n-subcategory which is replete. On the other hand there are two (three) DIFFERENT notions of image of a functor. One is the image in external sense, that is image in Cat taken as a category or as a 2-category. Another is image as a subcategory in literal sense. Image in literal sense is of course a very specific representative of an image in Cat 1-categorical sense and essential image is a very specifical choice of an image in 2-categorical sense; actually it is a specifical choice of such 2-categorical image that the embedding of essential image into the codomain is also literally surjective on objects. This is a bit strange from external point of view: you have something what is just equivalent to 1-categorical image, while it additional property is again of 1-categorical type. Thus it mixes the two. Hence it is by no means superimposable to homotopy limits in any case.

  • Urs Schreiber:

  • Andrew Stacey: Responded at paracompact space and Froelicher space. Incidentally, if the start of a query box is indented for some reason (as on paracompact space) then it seems that all its contained paragraphs should be indented by at least the same amount.

    • Thanks, Andrew. Maybe eventually somebody finds the time to move the insights gathered there out of the query box and distill them into proper entry text.
  • David Corfield:

  • David Roberts:

    • fixing up some statements at paracompact space. Added comment about existence partitions of unity being dependent on the category these will be constructed in.




  • Eric: Installed Cygwin so that I could convert Dugger’s Sheaves and Homotopy Theory from dvi to pdf. I uploaded the pdf to the nLab and added links to all references to the paper.

  • Zoran ?koda?: created comonad, added more on connection for coring and semifree dga. I think David’d confusion might be genuine: not to call with dash or not, that is easy question of exact synonyms, but rather how to cleanly separate DIFFERENT but similar notions of say biadjunction and pseudoadjunction; setups in which they appear: strict and nonstrict 2-categories and Gray categories; and kinds of (pseudo/2)-monads they induce…to mention a few. The Memoirs booklet by Tom Fiore and some papers by Lack, Marmolejo, Vitale, Kelly…may be useful to compare and decide in this regard.

  • David Corfield: Added a comment at free cocompletion, which got me looking for “pseudoadjunction”.

    I would trigger a new page for it, but don’t know optimal naming conventions.

    • Eric: Hi David. Now that we have redirects, you can feel less concerned about naming conventions. For example, if you start a page pseudoadjunction and people come out with pitchforks saying it should be pseudo-adjunction, we now have the capability to simply change the page name. Better than that, we can add redirects so that both pseudoadjunction and pseudo-adjunction point to the same page and then it doesn’t matter. People can use either one when linking to your page.

  • Tim: I have started an entry on dg-quiver. I have paused because I cannot decide whether this is the right version to put there or whether to use Peter May’s discussion in the talk that is linked to from that page.




  • Urs Schreiber:

    • Eric, why don’t you make the material on electromagnetism in media that you added into a proper section at electromagnetic field? Then we could move what is proper discussion between us into a query box, after all, while having the genuine material visible in the netry and not hidden in the section Discussion.

      • Eric: I’ve thought about this some more and something still bothers me about the idea. If electromagnetic properties, i.e. μ\mu, ϵ\epsilon, σ\sigma can be geometrically incorporated into the Hodge star via the metric, this implies: 1.) Maxwell’s equations in inhomogeneous media are wrong (although in vacuum they reduce to the familiar form) and 2.) that the Hodge star should involve convolution, i.e. the metric should have a memory. Has anyone put forward any serious theories of a “metric with memory”? Asking that questions give me a sense of deja vu (getting old sucks).
  • Eric:

  • Urs Schreiber:

    • added to twisted cohomology the May-Sigurdsson reference, mentioned their definition of twisted cohomology in terms of associated bundles of spectra and added a discussion on how that relates to the rest of the entry

    • it seems to me that linking to a page via a redirect has as a consequence that the linking page is not listed at the bottom of the linked page under Linked from . That’s too bad.

      • Eric: This is one of several issues that have arisen due to the new redirects feature that I would not worry about. In the future, this should work as desired. Redirects also produce unnecessary “Wanted Pages” on the “All Pages” page. See the forum.
    • added the reference to Abad and Crainic at Lie infinity-algebroid representation

    • realized only now that there is an entry semifree dga, so I added to that entry a remark on Lie infinity-algebroid and conversely added there a pointer to the former

    • replied at connection for a differential graded algebra, remarking that this seems to be essentially the structure discussed at Lie infinity-algebroid representation – this concept seems to be reinvented many times, just recently it seems that what Abad and Crainic describe in 0901.0319 is the same idea

    • if indeed calling recently revised should be avoided for the time being, it is all the more important that you indicate even small changes/additions here at latest changes

    • yes, I am relieved to see Mike back, if only temporarily, I was worried that the nnLab had lost one of its most valuable contributors

  • Toby Bartels:

    • Welcome back, Mike!
    • My experience is that Recently Revised, All Pages, and especially Search will degrade performance. I need them all, but I try to use them sparingly. Long pages (such as this one sometimes) can also degrade performance, but only temporarily.
  • Mike:

    • Replied to discussions at replete subcategory and pseudofunctor.

    • Sorry for suddenly disappearing; after graduating (thanks for the thoughts everyone!) I had immediate obligations to three coauthors that took priority, so I lost track of the nCommunity for a bit. (One of those papers, which people here might be interested in, should be appearing on the arXiv shortly.) Unfortunately I’ll now be traveling and out of touch for the next month, but then I’ll be back.

  • Urs Schreiber:

    • added three basic examples to metric space

    • filled more information provided by Todd Trimble into the entry paracompact space

    • created cup product

    • checked by private email with Todd Trimble and probably see my confusion at paracompact space now – replied there and added explicitly the example of second countable fin-dim manifolds

    • created magnetic current and electric current

    • I am getting the impression that the server runs much more smoothly when one avoids to call the “recently revised” page. This is a pity, because I used to go there all the time to see what’s happening, but it would be helpful to figure out if maybe the cause of the performance problems we see can be narrowed down further. Maybe calling “recently revised” causes the software to go through the entire database in an inefficient way.

      • Eric: When you mentioned this via email yesterday, I stopped viewing “Recently Revised” (which I had been viewing very frequently prior to that). The performance yesterday was MUCH better. It could very well be that performance is degraded when you view “Recently Revised” and possibly seriously degraded when several people attempt to access it simultaneously (which I’m sure happens frequently).
    • have a question at paracompact space concerning what it says there about the “long line” compared to what it says at locally compact space – this seems to be inconsistent to me


  • Urs Schreiber:

  • Zoran ?koda?: created Euler number (including Euler polynomial(s)) and expanded Legendre polynomial. Wasted part of the day browsing programming manuals about Ruby…interesting. Maybe something prompts me to be doing something about it :)

  • Urs Schreiber: created Karoubi K-theory

  • Andrew Stacey: Stumbled across the discussion at Timeline of category theory and related mathematics on bibliographies and realised that more people were keen for this to be sorted out than I’d thought. A few possibilities are laid out in the corresponding discussion on the forum. Please stop by and let us know what you want from a bibliography system so that we can design it according to what everyone wants rather than just what a few of us want.

    On that note, seeing as my mathematical skills are not in the mainstream of the current focus of the n-lab, I’m concentrating a bit more on technical support (stuff like the forum, bibliography, diagrams, useful little scripts like how to download the entire lab for offline browsing). There are lots of things that I (and the others who do a little hacking like this) could do but only so much time in which to do it. So if there’s something you’d like done, say a bibliography, that you think I could help with then the fact that someone actually wants it done pushes it up my priority list. However, unless you tell me about it or mention it somewhere that I will actually see it then I’m not going to do anything about it because I won’t know about it!

  • Urs Schreiber:

    • created Kalb-Ramond field

    • expanded the list of examples at model category and added at the beginning a sentence on combinatorial simplicial model categories

    • after a little reflection I moved the previous content at electromagnetism to electromagnetic field and kept just a brief note at the former, for later expansion

      then I worked on electromagnetic field

      I renamed the section I was working on into Mathematical model from physical input . This now starts with quick and concise derivation of the fact that the EM field is modeled by a Cech-Deligne cocycle based on a quick definition of Maxwell’s equations and the quantization condition.

      the following sections “the local picture” and “the global picture” are supposed to provide the remaining details and background. Still needs polishing.

  • Toby Bartels:

    • Fixed links at Timeline of category theory and related mathematics until I finally got tired (through 1969).
      • Then had to deal with the agonisngly slow server while I checked this.
      • And finally took a whole hour just to get access to this page so I could report on what I did —nothing else!
    • Having a discussion with Zoran ?koda? about transliteration at M M Postnikov.
    • Removed the redirect from Cauchy colimit to Cauchy complete category on the grounds that they are not actually discussed there.
      • Restored the link from direct sum accordingly (but maybe it should not link there?).
      • Changed the example at redirect accordingly.



  • Toby Bartels: Wrote sphere and pointed space to fill some gaps. The former has a reference to (yet unwritten) Whitehead's theorem with the provacative claim that this shows that generalised (Eilenberg?Steenrod) homotopy theory? is unnecessary; I don't really intend to defend that, but maybe it will interest the people working on that subject?

  • Andrew Stacey: Started a discussion on the n-forum about how to get a snapshot of the n-lab (since this is really an announcement page rather than a discussion page).

    Discussion is here.

  • Toby Bartels: Added some illustrations to simplicial set, based on those at cubical set, as requested by an InterestedAnonymousCoward.

  • Zoran ?koda? created microbundle. Note that classical references do not mention morphisms, just isomorphisms or equivalences of microbundles. Did anybody notice my update downstairs on the issue of export_html (answer to Urs/Toby answers) ? I suggested that once a week an export_html be posted as a file to be downloaded which is not up-to-date with a warning, as I think (maybe I should be corrected) that Jacques stopped serving export_html because of long generation/compilation time, while static file and new cimpilation once a week will do less harm. And leave generation of export_markup as it is, up-to-date.


  • Urs:

  • Toby Bartels:

    • Another tip for Zoran: If ab\sqrt{\frac{a}{b}} looks bad, then the problem is on your end. To be sure, it looks bad to me too, but that problem is on my end; it looks good if I use the STIX fonts (as discussed here), but I think that those are otherwise pretty ugly, so I don't use them. So the problem is that almost every font doesn't know how to do that sort of thing correctly … but the MathML produced by Instiki is correct. (Update: Actually, that example looks just fine in DejaVu Serif too, but I remember that there are other examples that don't.)
    • A question on terminology at replete subcategory. Possibly the answers should inform equivalence.


  • Urs:

  • Tim: I have started to reorganise some of the entries on Cech methods since David has started on homotopy (as an operation) and had an idea about Cech homotopy. I have encorporated a point made by Zoran? about the history of Cech methods.

  • Zoran ?koda?: created a version (to be expanded) of Legendre polynomial and M M Postnikov and added references to Postnikov system. I think historically tower and system differed by inclusion of universal cohomological class representing the fibration into the notion of system (cf. Whitehead’s big book, ch.9). This should be still noted: if one does not specify the cohomological class this is I think like missing the choice of isomorphism when the isomorphism exists. Technical issues: I encountered a problem that sqrt{fraction} puts sqrt only such that the numerator is under the root. I do not know how one should write correctly. Toby thank you for the tip for getting the SOURCE of old versions. I sometimes write some items partly motivated by need to have them for my students, and plan to incorporate something close to my version into student scripta. You are very knowledgable about wiki world. :) I was also trying to take the export of the whole nlab and succeeded to get the markupML version but not html: when asking nlab/export_html i get 403 FORBIDDEN message in my browser.

    • Urs Schreiber: I also get this error message when trying to export the nnLab as html – I remember the html export was particularly heavy on the server and maybe it was truned off in view of the server being a bit weak – we are trying to move to a better host eventually

    • Toby: Yes, Jacques turned that off because it was such a load on the server; I expect that we can turn it on again when we get a better host. In principle, you can get the HTML by exporting the source and compiling it on a local copy of Instiki, but of course you have to install Instiki to do that. (Also note that you'll need the CSS files if you want the HTML to look the same, including fonts, query boxes, etc.) And neither of these includes old versions; I think that Urs(?) is backing those up periodically in case the server crashes.

    • Zoran: could then somebody make one copy of html zip file 2-3 times a month ? It would not be updated but still it would be useful for browsing math when offline. If I get the zip-file I can put it online on my homepage. Or simply could Jacques put one zip file of export_html weekly with link and warning that it is not up-to-date; and for editing we can anywy use markup version. Then the server does not get heavy with generation, I think that probably generating, compiling all takes time, the shear downloading from time to time would maybe not burden the server ?

  • David: began an experiment on homotopy (as an operation) of dualizing the cohomology page. Began generalized (Eilenberg-Steenrod) homotopy.

  • Urs Schreiber:

    • rewrote the intro to Cech cohomology (see there) and started adding a list of examples for the abelian case: the standard series line bundles, line bundle gerbes, with connection, etc. – but not well written at the moment and no effort to get signs straight

    • created nerve and realization in order to host Kan’s general idea of how a functor into a category with colimits induces an adjoint pair of functors

      • I think I know what I am doing, but I’d like to ask people like Tim Porter and Todd Trimble to have a critical look at it (where is Mike Shulman??) – in particular at the moment I allowed myself to assume that we are copowered over the enriching category in order to get nice formulas, I wouldn’t object if somebody finds the time to give the more general discussion

      • then of course I adjusted links and made some comments accordingly at nerve, geometric realization and Dold-Kan correspondence

    • following Eric‘s question I typed a quick reply into cohomology on how the ordinary notion of cohomology in cochain complexes is reproduced. In principle this gives all the necessary information, but I’ll try to find the time later to give a long detailed exposition of how this basic important special case arises from the very general perspective

    • added an “Idea” section to the beginning of Postnikov system in triangulated category

  • Toby Bartels: A quick note: I also have been changing <span class='newWikiWord'>apple<a href='/nlab/new/apple'>?</a></span>s to <a class='existingWikiWord' href='/nlab/show/apples'>apples</a> but I will not do it now unless I have reason to think that it was written by someone who prefers <a class='existingWikiWord' href='/nlab/show/apples'>apples</a>. In general, I do not edit matters of taste; I didn't know that this was a matter of taste, but now I do know that. (Sometimes if I'm changing something else, then I will change matters of taste at the same time, but only if I have substantially rewritten the sentence or if helps to standardise the notation and terminology. And this does not qualify as notation and terminology.) I'm sorry if I caused offence, but please understand that I did not know that there was a difference of style.

    • PS: If Eric adds “o the bottom of <span class='newWikiWord'>apple<a href='/nlab/new/apple'>?</a></span>, then both and should work. I will still add such redirects myself, for the benefit of the style preferred by Urs, Eric, and me; but I will no longer change styles in what Zoran has written.
    • PPS: To see the source of an old version of a page, hit Back in time until you find the right version (or try History to get a list of versions), then hit Rollback to see the source of that version. After you've copied the source, hit Cancel (or Submit if you really want to change it back to the old version).
    • PPPS: You can type `apple?s` to get <span class='newWikiWord'>apple<a href='/nlab/new/apple'>?</a></span>s if you find it convenient to do so.


  • Zoran ?k?oda: but the plurals are NOT there – if I write [[apple]]s I did not use the code for plural. Let me clear the issue (I will write round brackets): Eric is doing TWO things 1) he is taking entry ((apple)) and adding the redirection instruction inside to allow for ((apples)). This creates one new version, not too bad, you consider this robust, I can tolerate it. 2) he is changing every occurence of my reference ((apple))s which used to be correct usage from within entries ((banana)), ((pear)), ((ananas)) and ((strawberry)) to new format ((apples)). This amounts to not allowing me to use legitimate ((apple))s from within ((bananas)). This second thing, unlike the first, I can not tolerate, as it has no rational explanation. I do not know if it is good :)).

    • Eric: The fact that many items appear as [[apple]]s on the nLab is an artifact of the period prior to having redirects. Prior to having redirects, we’d have to write that as [[apple|apples]] to get it to render correctly which gets old after a while, so people naturally gravitated to the easier [[apple]]s. If we’d had redirects from the beginning, there would be a redirect at [[apple]] for [[apples]] and no one in their right mind would ever write [[apple]]s again (which is distracting to look at) if they could just write [[apples]] instead. I’m at a total loss as to why you oppose this. Currently, I am trying to reverse the damage so that we can make things cleaner from here. Whenever, I see “]]s”, I instinctively change this to “s]]” and add a redirect if it doesn’t exist. In time, this should work itself out and we should have plural redirects for most links that are commonly used. It would work itself even faster if people stopped writing “]]s” and used the plural link form instead.

    • Zoran: I disagree that for example I write [[apple]]s rather than [[apple|apples]] beause it is easier. I write it because the appearance of [[apple]]s I like more: it tells me by the color which part of the name is real URL (as I often type URL), plus I have no dislike for multicolored words. I will continue writing like that. If you like to write your way write, but leave my links the way they are. Otherwise I can not ENJOY writing. It takes sometimes the whole minute to reload the page and I often reload the page is somebody has changes it, and it disappoints if the change is a matter of taste, and is legitimate. Second there may be day when you will have no time to write redirects etc. and one will not memorize which redirects exist and which do not. If I write the way I do, I will never have problem with this. If I want a single-color appearance of the link for some reason, I do not mind writing it long way like [[apple|apples]], it is about 2 seconds more, rather than spending far more time to check if there is a created redirect and wait half a minute to load the page, specially if in addition to slowness of the server I have my own internet connection problems what is very often.

  • Urs Schreiber

    • I started going a bit through the Timeline of category theory and related mathematics and added links to nnLab entries wherever I saw a possibility

      • in that context: I like what Eric is doing. It makes the linkage of the Lab more robust. For instance quite a few of the Timeline’s imported links didn’t break (while thousands broke) just because Eric had made sure that some plurals etc were recognized.
  • Zoran Skoda

    • created Postnikov system in triangulated category

    • I STRONGLY disagree with creation of spurios PLURAL items when they are of NO use. Namely like Eric just created new version of bialgebra cocycle to say that it redirects bialgebra cocycles, while it is simpler and better from memory point of view to write bialgebra cocycles. Why having whole page archived and one more page to browse in history just to distiguish if s is before or after the brackets ?? Dear Eric we have thousands of items to enter and there will be thousands of new pages in future and why to increase entropy and spend yuor valuable time on this – please take some book and help us entering something NEW and not messing with plurals and creating new versions for nothing. I have hard time browsing history when something is messed up and if I am going to spend it on such a nonsense than I will leave the idea to enter new items myself.

      • Eric: Hi Zoran. By adding a redirect for plurals, we are not creating spurious pages (but we are creating spurious revisions, but I don’t see that as a meaningful issue). In fact, my hope is that everyone will stop writing things like [[page]]s in favor of [[pages]] at which point I could stop correcting the links. Please see redirect for more information. Redirects are a very nice new feature of the nLab (partially motivated by your suggestions wrt symbolic links) and I hope they become a natural part of any contributors arsenal. PS: It is ironic that you complain about creating additional revisions when you just created a THIRD revision by changing my redirect BACK (???). PPS: A good place to discuss this and any other administrative issues is on the n-forum. That is a good place for such discussions and once a decision has been made about any issue, the conclusion will be placed on the brand new nLab meta site. We’re working on decision making processes now and any feedback is more than welcome.
    • Zoran: look at my explanation few paragraphs above: I accept your creating redirects, but do not accept not allowing me having my own format of calling links within my own text. I changed your redirect back because I want to assert my right to have the link called any possible way I like it. It functions, it correctly displays and it si even more informative: ((apple))s tells you even the information that s is not the name of the page, and I often TYPE the URL name of the page rather than clicking on the link, because often because of slowness of the server working on laptop and desktop simultaneously. Of course this kind of strange usage is useful just for the author, but if I am in process of improving the page which i largely created i think I have the right for the convenience.

      • Eric: Hi Zoran. The nLab is a group effort, and as such, it has to have some rules and should probably have at least some (loose) sense of uniformity. I don’t think it makes sense for Toby and I to be going around correcting links to have you change them back simply because you want assert yourself. The nLab is already MUCH looser than Wikipedia as far as standards and formats and I think that is a good thing, but I also think there should be some agreed upon “Matters of Style”. I started a discussion on “Matters of Style” on the n-forum here. We should probably discuss it there. Whatever we all decide on, then we can add a “Matters of Style” page to nLab meta. Anyone reading this is more than welcome to voice their opinion, but once a “quorum” is met, I think we should establish a rule. In a bit, we may also want to remove this discussion from latest changes? since this isn’t what the page was intended for.
    • created bialgebra cocycle, Drinfel'd twist

    • by the way how does one download the source of a previous version ? I sometimes create a page and then there are changes after and I want to have my file for other purposes with what I wrote and I do not know how to access it.

  • John

  • Urs

    • am working at Cech cohomology on the section “Abelian Cech cohomology” where my aim is to spell out a derivation of the standard Cech double complex from starting with the general definition of cohomology for the case that the coefficient object is in the image of the Dold-Kan map from chain complexes of sheaves to simplicial sheaves – I am not really satisfied, but this is how far I got – check critically



  • Toby Bartels:

  • David

  • Urs

  • Eric

    • Reminds Urs that he no longer needs to type [[Urs Schreiber|Urs]] because there is now a redirect from Urs to Urs Schreiber., i.e. the link [[Urs]] points to Urs. Try it! :)

    • Along similar lines, after seeing [[internalization|internal to]] about a million times, I just added a redirect so that typing [[internal to]] gets redirected automatically to internal to. Try it! :)


  • Urs

  • Toby Bartels:

    • I would like to suggest that the appearance of links to nonexistent pages is a feature that we should not break. Thus we should not create blank pages (or pages that are blank except for redirects) but instead create pages only when we have something to put there. Conversely, we shouldn't change links to go to redirected forms (as at geometric infinity-function theory currently) unless the redirects have actually been created. If this means that we have things like <span class='newWikiWord'>infinity-foo<a href='/nlab/new/%E2%88%9E-foo'>?</a></span> (when nobody has written about \infty-foos yet), then we're no worse off than before we had redirects, and the appearance of links to nonexistent pages still tells us something. (Note: there is some related discussion now hidden under July 3.)

      • Eric: I agree that appearance of links to nonexistent pages is a feature that should not be broken. Currently, one of my self-assigned projects (a.k.a. a labor of love) is to pick a page and systematically “clean it”. The last major cleaning was geometric ∞-function theory. I hope that for the most part (with some minor exceptions) most everyone would agree that it looks a lot better now. The issue that brought up your comment, however, came about as a result of me “cleaning” strict ∞-category. Part of my system for cleaning pages involves changing things like [[category|categories]] to simply [[categories]] and adding a redirect from categories to category. If I do this enough, then most n-Lab pages will have available redirects for plural forms of nouns. In fact, I would encourage everyone to stop including things like [[singular page name|plural page name]] and simply use the plural form [[plural page name]]. Then either leave the nonexistent link there for someone to add the redirect or add the redirect yourself. What happened on strict ∞-category was that there was a link [[exchange law|exchange laws]]. My “system” suggests that I change that to [[exchange laws]] and add a redirect from exchange laws to exchange law. In this rare case, exchange law did not exist yet, so I couldn’t add a redirect without creating a blank page for exchange law. In this rare case, given that I am trying to systematically “clean” pages, I thought it was acceptable to break the rule about nonexistent pages. I still think it is justified as long as it doesn’t become a habit, which I’m pretty sure it won’t be.
    • A question at cogroup about what we really want there. (Surely more than just the empty set?)

      • Urs: we want to say that pointed spheres are co-groups, so that maps out of them, called homotopy groups, are, well, groups. Supposed to be dual to the statement that mapping into a group coefficient object gives cohomology groups
  • Urs

    • added to principal bundle a long detailed section called “the GG-action from the homotopy pullback” – this may look like weird overkill, but the point is that this serves as a warmup for an analogous discussion at principal infinity-bundle

    • rediscovered that we had an entry Cech methods and added lots of links to that

    • provided explicit details at Cech cohomology for the general (nonabelian) case in low degree

  • Zoran ?koda?: created small fibration, added more general discussion on endomorphism monoids.

  • Urs created Cech cohomology

  • David

    • more suggestions than changes, but it would be good to have entries for cogroup and co-H-space. Could homotopy group and cohomology group be made to resemble each other more? I.e., must the former be restricted to nn-spheres as domain? Hmm, is something suboptimal about H-group and H-cogroup, whereas H-space and co-H-space? Perhaps ‘co’ and ‘H’ commute.

      • Urs: it would in principle be good to have expositions at cohomology, cohomology group and homotopy, homotopy group be more symmetric – we just have to beware that we’d be going into untrodden territory and should indicate accordingly: while generalized cohomologies of all sorts are becoming familiar, I can’t recall having seen mentioned the corresponding dual notion of generalized homotopy anywhere but in our discussion – so we should maybe tag the corresponding discussion, if we implrement it, with a box saying “this is research material” or the like – but I would enjoy it if we did so
  • Urs





  • Urs

    • started a section “concrete realizations” at principal infinity-bundle. So far I recall the old result by Quillen on certain “1-categorica topological bundles” and their \infty-action groupoids. Then I start making some remarks on Jardine’s approach using what he calls “diagrams” and have a remark on how that compares to the Bartels-Baković-etc-style. This requires eventually much more discussion, but I have to call it quits for today.

      But the point is that Jardine works in the “petit topos” perspective where all bundles live over the fixed site. So the terminal object in his setup is not the point, but the underlying space over which one works. This means that the simple picture of a principal \infty-bundle as the homotopy pullback of the point no longer works and is the reason why he introduces the yoga of what he calls “diagrams”.

      On the other hand, when one works with simplicial sheaves in the context of a “gros topos” such as sheaves on DiffDiff or on open subsets of n\mathbb{R}^n or equivalent, then the simple conceptual picture remains valid.

      Notice that placing oneself into the context of nn-groupoids internal to diffeological spaces or the like is doing precissely this: working with simplicial sheaves on DiffDiff.

      Evidently i shouldn’t be discussing this here but in some entry. But it’s time for me to go to bed now.

  • Zoran ?koda? created grouplike element. It contains few words on Amitsur complex for a coring with a (semi-)grouplike. The aim is to soon (using the setup) introduce entries for connections for corings; and then correpsondence between falt connections and descent data in the comonadic and coring setups (after Menini et al; all coming back to the example which is the correspondence between 1-order costratifications and flat connections in the crystalline setup due Grothendieck).

  • Urs

  • Zoran ?koda? created coseparable coring,Sweedler coring, two dimensional sheaf theory; expanded stratifold (which was empty, but existing!), added a reference to fibration in a 2-category and somewhere else. I think that in K-theory delooping has a bit different multiple meanings which are related but are more procedures making from something what can not be delooped strictly in the sense of delooping to the delooping of something what is the best approximation of deloopable; there are procedures due to Quillen, Waldhausen, Karoubi etc.

    • Urs: the entry on two dimensional sheaf theory is motivated from some behind-the-scenes discussion Zoran and I are having – Zoran rightly points out that the present characterization of derived stack may be wanting, as strictly speaking saying “derived stack” should be related to but not be regarded as equivalent to “higher stack on higher categorical site” – what we really need eventually is more details on the Toën-Vezzosi work on higher topos theory in the context of SSet-enriched categories,
  • Urs

  • Eric: Added a redirect for Urs so that he no longer has to type [[Urs Schreiber|Urs]] and can simply type [[Urs]] and will be redirected to the correct page.


  • Urs

    • proudly presenting what is now the widest (horizontally speaking) nnLab entry as yet: at model structure on simplicial presheaves I added a section “Map” where I draw a big diagram that indicates at least part of the collection of model structures, their interrelation, definition and authors

      • Yeah, I was thinking of reworking that map to run vertically …. —Toby

      • Would that really be better, though? Optimally, eventually we’d produce a LaTeXed pdf diagram. Here and elsewhere. That, however, inhibits joint editing a bit. —Urs

    • replied at delooping, at group, added a bit at Notation?

    • moved the key statement of Toby‘s remark to the very beginning of group

  • Toby Bartels:

    • Andrew Stacey may be interested to see how I've changed the formatting of the goal box at Froelicher space.
    • Incorporated Zoran's new references at principal 2-bundle into the text.
    • Added a note at group how GrpGrp is a full sub-22-category of Grpd *Grpd_* (even though not of GrpdGrpd).
  • Urs

  • Toby Bartels:

    • I archived the 2009 June changes by the sneaky method of changing this page's name, copying the header and footer text (with appropriate changes) from the previous archive, and then creating this page anew. Maybe that will keep us from testing Instiki's tolerance for pages with thousands of edits in their history. (^_^)
    • I've completed migrating all old redirect pages to history pages; all links within the body of the Lab should now take you immediately to the target page. (There are a few victims of a bug, but I'll straighten that out with Jacques.) I have more comments on the Forum.

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