nLab 2009 June changes

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







  • Todd: Recently loop space was created. Also, I added some examples to Chu construction and to star-autonomous category.

  • Urs:

    • added the example of principal bundles to fibration sequence

    • added an example section to group cohomology with details of how the abstract-nonsense definition reproduces the familiar formulas

    • created twisted cohomology

    • created n-group

    • created group cohomology

      • the point of view adopted there is almost fully explicit in Brown-Higgins-Sivera’s book Nonabelian algebraic topology , only that one needs to notice in addition that the morphisms out of free resolutions of crossed complexes discusses there are the anafunctors that compute the morphisms in the corresponding homotopy category. Has this been made fully explicit anywhere in the existing literature?
    • fiddled a bit with the entry higher category theory (added one more introductory sentence, created a hyperlinked list of definitions of higher categories) but I still feel that we should put more energy in this particular entry. It is sort of the single central entry one would expect an “nnLab” to be built around, but currently it doesn’t even come close to living up to playing such a pivotal role. I am imagining that it should carry some paragraphs that highlight the powerful recent developments in view of Pursuing Stacks, of the kind that I filled in today in the entry Carlos Simpson. Does any higher category theory expert out there feel like writing an expositional piece for the nnLab here?

    • created simplicial localization but was then too lazy to draw the hammock. But main point here is the link to an article by Tim Porter that nicely collects all the relevant definitions and references

    • created principal infinity-bundle, just for completeness

    • created principal 2-bundle – this is just the result of what came to mind while typing, I am sure to have forgotten and misrepresented crucial aspects. Toby Bartels should please have a critical look and modify as necessary, as should Igor Bakovic and Christoph Wockel in case they are reading this.

    • based on a reaction I received concerning my comment below on the entry gerbe I have split the material into entries gerbe (as a stack), gerbe (general idea), bundle gerbe and kept at gerbe only pointers to these entries – let me know what you think

    • created an entry for Carlos Simpson motivated by a link to a recent pdf note – that Zoran Skoda kindly pointed me to – where Simpson briefly sketches the topic of higher stacks, old and recent progress and putting his own contribution into context. I thought that was a nice short comprehensive collection of keywords and so I reproduce that text now at the entry Carlos Simpson with all the keywords hyperlinked

    • created fibration sequence

    • added an “Idea”-section to gerbe supposed to be read as “general idea”, where I try to describe the concept in a way independent of the notion of stack, relating it to princpal bundles, principal \infty-bundles, fibration sequences and cohomology . At the end I say “in the following we spell out concrete realization of this idea”. Then I made Tim Porter‘s material a section “Realization of gerbes as stacks”. Eventually I’d like to add similarly “Realization of gerbes as Stasheff-Wirth fibrations”, “Realization of gerbes as bundle gerbes” etc.

      • I hope that this is okay with you all, in particular I hope that Tim doesn’t mind me putting such a chunk of material in front of his work. If anyone has the impression that the chunk I added should rather be separated to a different entry, I’ll have no objections. But please have a look first to see what I am trying to get at.
  • Tim:

    • I have added material to gerbe, although it is still a long way from explaining the link with cohomology with integer coefficients that was requested.

    • In the process of doing the above I have added a deconstruction section to torsor, and created trivial torsor.

    • I created a brief entry on Jean-Luc Brylinski, but this is really a stub with a link to Wikipedia.


  • Finn Lawler: Added a proof (sketch) of the ‘lax Yoneda lemma’ to lax natural transformation. Also replied re terminology at modification (thanks to Urs for a fantastic reply to my question there).

  • Zoran Škoda: created Hochschild-Serre spectral sequence.

  • Eric: I’ve started the process of applying redirects for symbolic links. This maintains the original ascii titles and urls, but makes links look much better. I’ve also started adding redirects for plural nouns, e.g. you no longer have to type [[category|categories]]. Now you can simply type [[categories]] and will be redirected automatically to category. To see the changes in action, have a look at higher category theory. Feedback welcome!

  • Zoran Škoda: Created Hopf envelope, free Hopf algebra and matrix Hopf algebra simultaneously with posting a related comment to a discussion between Baez, Trimble and Vicary on free and cofree functors for bialgebras.

  • Tim: I have created a stub on gerbes. There was a request on the Café for en entry and I thought this was a good way to remind me to start one. … but feel free to do it for me!

  • Zoran Škoda:

    • Created class of adapted objects. It would be better to say class of objects adapted to a functor (we talk half exact additive functors between abelian categories). Created Grothendieck spectral sequence and spectral sequence. Both need more input/work…

    • I have added a general paragraph at the beginning of descent because far not all the cases of descent theory fit into the framework of sheaf/stack theory, Grothendieck topologies and homotopical methods. Namely something is a sheaf when it satisfies the descent for all covers, but there are cases when one considers just the descent problem for a single morphism, not all morphisms of anything like a Grothendieck topology. There is also a descent along families of noncommutative localizations, when one needs to resort to noncommutative generalizations of Grothendieck topologies. So descent theory is more general than the geometry of sheaves and stacks, though the latter is surely he most important part.

  • Urs:

  • Toby Bartels: Just more questions, instead of answers, for Finn and Gavin.


  • Gavin Wraith wrote matrix theory, tensor product theory, and bimodel, with a question at the last.

  • Finn Lawler: created modification (and asked a question there) and lax natural transformation, including a statement of what I think is the Yoneda lemma for them. I have a proof (I think), which I’ll add a sketch of later, if no-one finds the statement obviously wrong in the meantime.

  • Urs

    • added to weak factorization system the full details of the (elementary) proof that morphisms defined by a right lifting property are stable under pullback –

    • added the theorem to Kan fibration that Kan fibrations between Kan complexes that are also weak equivalences are precisely those with the right lifting property with respect to simplex boundary inclusions

    • added example of fundamental \infty-groupoid = singular simplicial complex to Kan complex

    • edited Kan complex slightly and added two little propositions characterizing groupoids in terms of their Kan complex nerves

    • added further illustrations to Kan fibration


  • Toby Bartels: A section on measurable space about the constructive theory due to Cheng. (This is not a straightforward variation on the classical theory but distinctly a generalisation, so may be of interest even to classical mathematicians.)

  • Urs linked to normal complex of groups by a little lemma now added to Moore complex

  • Tim Porter added normal complex of groups

  • Urs:

    • worked on simplicial group:

      • added section about the adjunction between simplicial groups and simplicial sets, in particular mentioning the free simplicial abelian group functor (since that plays a role in the proof at abelian sheaf cohomology, which now links to it)

      • moved statement and proof that every simplicial group is a Kan complex into a formal theorem-proof environment

      • added references to Goerss-Jardine, to Tim Porter‘s Crossed Menagerie and to Peter May’s book “Simplicial objects in algebraic topology”

      • added an “Idea” section

    • added to Moore complex

      • the theorem of how it relates to the other two complexes in the game (it is quasi-isomorphic to the alternating sum complex and isomorphic to that divided by degeneracies)

      • illustrations of cells in low degree

      • the reference to Tim Porter‘s The Crossed Menagerie notes

      • a brief comment on use of terminology (after checking with Tim Porter)

  • Toby Bartels: Started more articles that I'd linked earlier: exclusive disjunction (with a philosophical claim that I need to back up), countable choice (very much a stub), sigma-ideal, Hartog's number.

  • Urs:

    • added the detailed formal proof to abelian sheaf cohomology that shows how it is sitting as a special case inside the more conceptual nonabelian cohomology (= hom-sets of infinity-stacks, really)

      • in this context I also rewrote the “Idea” section in the desire to drive home the main point better, but without removing the previous material. As a result the text of the entry is now somewhat repetitive, I am afraid. Maybe a little later, when I have more distance to it, I’ll try to trim it down again. Or else, maybe one of you feels like polishing it.
  • Toby Bartels: Wrote Moore closure, since I linked it and it's a neat idea. As I was writing it, I realised that it was more abstract than what I'd been writing lately and must correspond to something very nice categorially; by the time I was done, I realised what it was: a special case of a monad. Then I saw that there were really no examples of monads on our page!, so I added a few, but the majority of examples of monads on the wiki now are Moore closures. (And believe me, Moore closures are everywhere.) Anyway, if you're trying to understand what monads are, why not try Moore closures first?

  • Urs

    • added an “Idea” section to Moore complex, edited the section headers a bit – and have a question onm terminology: isn’t this really the “normalized chain complex” whereas the Moore complex is the one on all cells with differential the alternating sum of face maps?

    • created organization of the nLab? – please see there for what this is about (or in fact, see the corresponding nForum thread that is being linked to there)

    • added references to Goerss-Jardine’s Simplicial homotopy theory to Dold-Kan correspondence and Moore complex

    • reacted a bit at An Exercise in Kantization – behind the scenes this is being developed further, I’d be happy to provide more detailed replies in a while, when things have stabilized a bit more

    • added a mention of and a link to Kan lift to the “Idea” section of Kan extension

    • replied to the discussion about pullback notation at Kan extension: I originally had “p *p^*” there. After somebody changed that to “p *p_*” I wrote the section “note on terminology”. I’d be happy to have the p *p^* reinstalled.


  • Toby Bartels:

  • Todd:

    • I have a notational comment at Kan extension. Spurred by David Corfield‘s post at the Café, I hope to get started on Kan lift soon. (Done.)

    • There’s a running discussion between Toby Bartels and me at cyclic order, centering on whether Connes’ cycle category Λ\Lambda has been correctly characterized, or if not how to fix it. We agree now that a notion of “total cyclic order” is classically equivalent to the “linear cyclic order” notion used in the article, and equally feasible for purposes of trying to characterize Λ\Lambda, but I’m currently perplexed by the apparent presence of a terminal object.


  • Tim: Finally I have created 2-crossed complex, and in the process needed to create normal complex of groups. What would be nice is to work out exactly what \infty/ω\omega-groupoids correspond to 2-crossed complexes. Any ideas? (It probably is obvious viewed from the right perspective but I fear I do not yet have the right perspective to say ‘aha!’) I can characterise these objects in homotopy theoretic language, but really would like some neat way of describing them in \infty-cat terms.

  • Todd Trimble: asked a question of Mike and Toby over at cyclic order.

  • Toby Bartels:





  • Toby Bartels:

    • Wrote topological category, including a lot of vague stuff there, since I don't have good references here now. But I'm pretty sure that everything that I said is at least true.
    • Wrote core, using a term that I learned from Mike Shulman for an underlying groupoid.
    • Wrote semigroup and magma, since they're such basic topics, but I didn't say much.
    • Added stuff to reflective subcategory about how obects of the ambient category can be seen as objects of the subcategory equipped with extra structure. Besides the examples there, see also core.
    • In reply to the last point in what Urs has said immediately below, if you want to put in something that you expect to be controversial, and hesitate for that reason, then put it in and just label it as controversial. Put in a query box asking ‘Is this right?’ or saying ‘This is my new definition.’ or whatever. Or put it in a new section at the bottom of the page, marked ‘## Uncertain material’ or whatever. The point is, get it down one way or another, and we will be very happy!
  • Urs: I have a comment and appeal at nInsights

    • in this context I also want to ask again everybody:

      • please don’t forget to drop latest changes logs here alerting us of which changes you made where, if it’s anything beyond fixing typos

      • if you feel something needs to have a place in the nLab, don’t feel hesitant to add it. I am getting the impression there is in parts some hesitation to insert material without checking with everybody else first. I’d say more efficient would be: add the material you would like to see, and if it should really turn out that there is serious disagreement with some other contributor, we can still roll back things and look for compromises.

  • Urs:

    • expanded inverse image: added an “Idea” section, restructured discussion into presheaf and sheaf and sheaf on top-space bits, added the crucial lemmas and theorems and provided detailed proofs of some of them.

    • replied again at simplicial homotopy group: Tim or Toby should please feel free to implement the remaining terminology adjustment

    • reacted and replied at simplicial homotopy group (also have a question) and added a few further bits

  • Tim:










  • John: I created myself a new personal web since I lost the password for my old one — and besides, the name of my old one was nonstandard. So far the only thrilling feature of this new web is the introduction to a paper I’m writing with James Dolan, tentatively titled ‘Doctrines of Algebraic Geometry’.

  • Mike: Tried to distill a bit of the cafe discussion about the empty space.



  • Toby Bartels:

    • Thanks for prometric space, Mike. I didn't know about those, but they seem quite reasonable. I was thinking that I might want to figure out the categorial meaning of the definition of gauge space, but maybe it's just that it's a halfway attempt at prometric space! (But do you have any good example of a nongaugeable prometric space?)
    • I seem to have gotten into an edit conflict with you at Zorn's lemma, Todd. I didn't tell it to override your edit, but something happened regardless. Anyway, I think that I've fixed it.
    • Also, our page is at Hausdorff maximal principle, which seems to be more common, but maybe your name is better. But look!, now it redirects! And you can move it too!
  • Todd Trimble: gave a proof of Zorn's lemma. Wouldn’t mind expanding that entry to include the mutual equivalence between AC, Zorn, and well-ordering principle (assuming excluded middle). May get around to putting in something at Hausdorff maximality principle.

  • Mike: Inspired by gauge space, created prometric space.

  • Toby Bartels:

    • Eric and I have been testing the new move and redirect features at the top (for some reason) of the Sandbox. It seems possible to move and to create new redirects (make sure that you see how, it's backwards from MediaWiki), but not (yet) possible to regularise all of the current redirect pages.
    • Created a few requested pages. The interesting ones are Bill Lawvere, Boolean ring, and ideal.



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Last revised on August 24, 2015 at 02:30:21. See the history of this page for a list of all contributions to it.