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    • A list of all recently edited entries can be seen at Recently Revised. But that list tends to contain lots of minor changes: it's not easy to spot the important ones. So, if you feel people's attention should be drawn to some changes you make, please mention them here. This way the rest of us can spot them, so we can learn what you know --- and maybe make further improvements!

      1. Go to the Latest Changes category in the Forum. If you are reading this on the forum, then you are there right now, or go to http://www.math.ntnu.no/~stacey/Vanilla/nForum/?CategoryID=5.
      2. Optionally, create an account (if you don't have one) and log in if necessary; this may be helpful but is not required.
      3. Hit ‘Start new discussion’ at the top left.
      4. Choose a meaningful subject title, perhaps the name of the page you've edited or something like that.
      5. Write a brief message summarising your edit. Most of the formatting is the same as on the Lab if you choose the ‘Markdown’ radio button beneath the comment box.
      6. Please include a [[wikilink]] to the page or pages that you edited.
      7. If you didn't log in, then please also include your name.
      8. Check Preview Post (or skip it), and then Start your discussion!

      Questions? Ask them here!

    • (edit: typo in the headline: meant is "bare" path oo-groupoid)

      I think I have the proof that when the structured path oo-groupoid of an oo-stack oo-topos exists, as I use on my pages for differential nonabelian cohomology, then its global sections/evaluation on the point yields the bare path oo-groupoid functor, left adjoint to the formation of constant oo-stacks.

      A sketch of the proof is now here.

      Recall that this goes along with the discussion at locally constant infinity-stack and homotopy group of an infinity-stack.

      P.S.

      Am in a rush, will get back to the other discussion here that are waiting for my replies a little later. Just wanted to et this here out of the way

    • created Tannaka duality

      with a short proof of the duality for the category of permutation representations of a group, using the Yoneda lemma three or four times in a row and nothing else.

      either I am mixed up (in which case we'll roll back), or I guess this is the way that it's usually done in the literature? I haven't really checked. Sorry, I just needed that quickly as a lemma for my discussion at homotopy group of an infinity-stack

    • For the record, all I did at geometric morphism#sheaftopoi was to add a paragraph at the beginning of the example, substitute ‘sober’ for ‘Hausdorff’ in appropriate places, and add to the query box there. I mention this because the diff thinks that I did much more than that, and I don't want anybody to waste time looking for such changes!

      I still to make the proof apply directly to sober spaces; the part that used that the space was Hausdorff is still in those terms.

    • I fiddled a bit with direct image, but maybe didn't end up doing anything of real value...

    • I have just uploaded a new 10 chapter version of the menagerie notes. It can be got at via my n-lab page then to my personal n-lab page and follow the link.

    • I've started a section in the HowTo on the new SVG-editor.

    • I thought I'd amuse myself with creating a succinct list of all the useful structures that we have canonically in an (oo,1)-topos without any intervention by hand:

      • principal oo-bundles, covering oo-bundles, oo-vector-bundles, fundamental groupoid, flat cohomology, deRham cohomology, Chern character, differential cohomology.

      I started typing that at structures in a gros (oo,1)-topos on my personal web.

      I think this gives a quite remarkable story of pure abstract nonsense. None of this is created "by man" in a way. It all just exists.

      Certainly my list needs lots of improvements. But I am too tired now. I thought I'd share this anyway now. Comments are welcome.

      Main point missing in the list currently is the free loop space object, Hochschild cohomology and Domenico's proposal to define the Chern character along that route. I am still puzzled by how exactly the derived loop space should interact with \Pi^{inf}(X) and \Pi(X).

    • I added a new section at curvature about the classical notion of curvature and renamed the idea section into Modern generalized ideas of curvature. The classical notion has to do with bending in a space, measured in some metrics. I wrote some story about it and moved the short mention of it in previous version into that first section on classical curvature. It should be beefed up with more details. I corrected the incorrect statement in the previous version that the curvature on fiber bundles generalized the Gaussian curvature. That is not true, the Gaussian curvature is the PRODUCT of the eigenvalues of the curvature operator, rather than a 2-form. Having said that, I wrote the entry from memory and I might have introduced new errors. Please check.

    • Todd suggested an excellent rewording in the definition of horn, and I have made the necessary changes. Do check it out if you care about horns!

    • I noted the point made by, I think, Toby about there being stuff on profinite homotopy type in the wrong place (profinite group). I have started up a new entry on profinite homotopy types, but am feeling that it needs some more input of ideas, so help please.

    • I added to covering space a section In terms of homotopy fibers that explains the universal covering space as the homotopy fiber/principal oo-bundle classified by the cocycle that is the constant path inclusion X \to \Pi_1(X) of topological groupoids.

      To fit this into the entry, I added some new sections and restructured slightly. Todd and David should please have a look.

      What I just added is essentially what David Roberts says in various query boxes, notably in what is currently the last query box. Back then we talked about the "Roberts-Schreiber construction" or whatnot, but really what this is is just the standard way to compute homotopy fibers in the oo-category of oo-groupoids.

      I suspect that Todd's bar construction described there can similarly be understood as being nothing but another way to compute the more abstractly defined homotopy pullback in concrete terms. I'll have to think about this, though. But probably Tim Porter or Mike Shulman will immediately recognize this as the relevant bar construction of homotopy pullbacks in homotopy coherent category theory.

    • edited Grothendieck's Galois theory

      a bit, added hyperlinks here and there, in particular linked to homotopy group of an infinity-stack -- also referenced the chapter in Johnstone's book.

      It would be good if we could highlight what exactly the theorem described there actually says. Currently it is easy to miss for the reader what the punchline is. But I have to do something else right now...

    • I am expanding the entry homotopy group (of an infinity-stack) by putting in one previously missing aspect:

      there are two different notions of homotopy groups of oo-stacks, or of objects in an (oo,1)-topos, in general

      • the "categorical" homotopy groups

      • the "geometric" homotopy groups.

      See there for details. This can be seen by hand in same cases That this follows from very general nonsense was pointed out to me by Richard Williamson, a PhD student from Oxford (see credits given there). The basic idea for 1-sheaves is Grothendieck's, for oo-stacks on topological spaces it has been clarified by Toen.

      While writing what I have so far (which I will probably rewrite now) I noticed that the whole story here is actually nothing but an incarnation of Tannak-Krein reconstruction! I think.

      It boils down to this statement, I think:

      IF we already know what the fundamental oo-groupoid \Pi(X) of an object X is, then we know that a "locally constant oo-stack" with finite fibers is nothing but a flat oo-bundle, namely a morphism \Pi(X) \to \infty FinGrpd (think about it for n=1, where it is a very familiar statement). The collectin of all these is nothing but the representation category (on finite o-groupoids)

      Rep( \Pi(X)) := Func(\Pi(X), Fin \infty Grpd)

      For each point x \in X this comes with the evident forgetful funtor

      x_* : Rep(\Pi(X)) \to Fin \infty Grpd

      that picks the object that we are representing on.

      Now, Tannaka-Krein reconstruction suggests that we can reconstruct \Pi(X) as the automorphisms of the functor.

      And that's precisely what happens. This way we can find \Pi(X) from just knowing "locally constant oo-stacks" on X, i.e. from known flat oo-bundles with finite fibers on X.

      And this is exactly what is well known for the n=1 case, and what Toen shows for oo-stacks on Top.

    • I wrote an entry (short for now) separable algebra. It is a sort of support for the current Galois theory/Tannakian reconstruction/covering space/monodromy interest of Urs.

    • These are used by Sridhar Ramesh to great expository effect at (n,r)-category.

    • in reply to Jim's question over on the blog, I was looking for a free spot on the nLab where I could write some general motivating remarks on the point of "derived geometry".

      I then noticed that the entry higher geometry had been effectively empty. So I wrote there now an "Idea"-section and then another section specifically devoted to the idea of derived geometry.

      (@Zoran: in similar previous cases we used to have a quarrel afterwards on to which extent Lurie's perspective incorporates or not other people's approaches. I tied to comment on that and make it clear as far as I understand it, but please feel free to add more of a different point of view.)

    • I'd like to create an entry gauge fixing. the usual method I use to create new entries (typing their name in the search form) does not work, since search redirects me to examples for Lagrangian BV. now I'm trying to put a link here to see if this works (but I'm skeptic..). should it not work, how do I create the new entry?

      thanks

      edit: it works!!!! :-)

      I'll now start writing the entry

    • I am a bit unhappy with the present state of local system. This entry is lacking the good systematic nPOV story that would hold it together.

      (For instance at some point a local system is defined to be a locally constant sheaf with values in vector spaces . This is something a secret blogger would do, but not worthy of an nLab. Certainly that's in practice an important specia case, but still just a very special case).

      But I don*t just want to restructure the entry without getting some feedback first. So I added now at the very end a section

      A general picture

      To my mind this should become, with due comments not to scare the 0-category theorists away, the second section after a short and to-the-point Idea section. The current Idea section is too long (I guess I wrote it! :-)

      Give me some feedback please. If I see essential agreement, I will take care and polish the entry a bit, accordingly.

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