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    • Added a minor alternative rendition of the definition of natural transformation. I'm not sure that it's particularly useful for anything, but it is at least true, with some aesthetically pleasing qualities.
    • 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 am re-reading Simpson/Teleman's "de Rham theorem for oo-stacks" and realize what I missed on first reading:

      they have essentually the statement that I produced recently, that oo-stacks on a site with "geometrically contractible" objects form a localy contractible (infinity,1)-topos in that the terminal geometric morphism is essential.

      They say this (slightly implicitly) at the level of homotopy cateories on page 5.

      I briefly recorded this in a somewhat stubby way now at locally contracible (oo,1)-topos - Examples

    • I have created a brief entry for Phil Higgins. Clearly more could be added for instance his work on using groupoids to prove group theoretic results.

    • I have started an entry on the decalage functor.

    • I expanded the beginning of physicscontents and edited a bit to bring out more structure.

    • Created solid functor with an SVG graphic. The SVG editor is awesome! Even if it is still a little buggy.

    • Jim Stasheff edited a bit at A-infinity category -- but did not yet work in the comments that he posted to the blog

    • I worked on improving (hopefully) and further expanding (a bit) the discussion of geometric homotopy groups at

      I give at

      • Idea now just the brief reminder of the simple situation for 1-toposes as described at locally connected topos;

      • and then at Defintion I state the very obvious and simple generalization of this to homotopy oo-groupoids of objects in a locally contractible (oo,1)-topos.

      Then I say something like: while this definition is very obvious and simple, it seems it has not quite been stated in the literature (except possibly in the thesis by Richad Williamson), but that there are old well-known results in the literature that essentially, with only slight modification of language, already do say precisely this.

      Then I go through this claim in detail. I list three subsections with three different methods of how to construct that left adjoint  \Pi(-) to the constant oo-stack functor, and then discuss in some detail how old and new references do already -- if slightly implicitly -- discuss precisely this. The three sections are

      You'll notice that I also link to the discussioon of the absgtract oo-adjunction on my personal web. Currently I am thinking of the entry on my personal web as talking about the abstract notion of a path oo-groupoid, and of this page here on the nLab as providing all the "well-known" aspects of it (in that these are in the literature).

      Please complain if somehow this doesn't look like the right thing to do. I am currently a bit undecided as to what bits of this discussion should be on my personal web and which on the nLab.

    • At category with weak equivalences we say that it is unclear whether every (oo,1)-category arises as the simplicial localization of a cat with weak equivalences, but that it seems plausible.

      At (infinity,1)-category we say that indeed every (oo,1)-category arises as the simplicial localization of a homotopical category.

      I had put in the paragraph that says this based on a message that Andre Joyal recently posted to the CatTheory mailing list.

      It would be good to harmonize this with the discussion at category with weak equivalences and maybe to add some references.

    • I think I understand locally contractible (oo,1)-toposes now, with their left-adjoint to the constant oo-sheaf functor.

      The (simple) observation is here on my personal web.

    • (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.


      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

    • 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 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.

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

    • 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?


      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.

    • I would like to write an article at size issues about the various ways of dealing with them, and I've started linking to that page in anticipation.

      I still haven't written it, but I decided that there were enough links that I ought to put something. So now there are some links to other articles in the Lab on the subject.

    • I noticed the word codifferential was used in the page on chain complexes. I raised this sort of terminological problem before and cannot remember the result of the discussion! The boundary operator in a chain complex is classically called the `differential', and the extra co seems contrary to `tradition'. Perhaps cochain complexes should have a cofferential but even that seems unnecessary. I have not changed this in case someone else has a good reason for the terminology ... what is the concensus? The point is not important but is worth clearing up I think.
    • Somebody clicked some buttons to make an empty slideshow at essential supremum, so I wrote an extremely stubby article there with links to real articles on the rest of the WWW.

    • I started subframe; of course, a subframe corresponds to a quotient locale.

      But we really want regular subframes. Is there a convenient elementary description of those?

    • Found some exposition by Todd on the web to add to ultrafilter, and a quote by Michal Barr added at ultraproduct.

      This was prompted by a comment by Terry Tao

      >There are two main facts that makes ultralimit analysis powerful. The first is that one can take ultralimits of arbitrary sequences of objects, as opposed to more traditional tools such as metric completions, which only allow one to take limits of Cauchy sequences of objects.

      So are ultralimits in his sense a form of completion?
    • Definitions for topological spaces, locales, and toposes here: open map.

    • Does anybody know of a definition of CABA (which would not literally be complete atomic Boolean algebra) such that the theorem that the CABAs are (up to isomorphism of posets) precisely the power sets is constructively valid?

      If you do, you can put it here: CABA.

    • I further worked on the Idea-section at cohomology, expanding and polishing here and there.

      Hit "see changes" to see what I did, precisely: changes (additions, mostly) concern mainly the part on nonabelian cohomology, and then at the end the part about twisted and differential cohomology.

      by the way, what's your all opinion about that big inset by Jim Stasheff following the Idea section, ended by Toby's query box? I think we should remove this.

      This is really essentially something I once wrote on my private web. Jim added some sentences to the first paragraph. Possibly he even meant to add this to my private web and by accident put it on the main Lab. In any case, that part is not realy fitting well with the flow of the entry (which needs improvement in itself) and most of the information is repeated anyway. I am pretty sure Jim won't mind if we essentially remove this. We could keep a paragraph that amplifies the situation in Top a bit, in an Examples-section.

      What do you think?

    • This comment is invalid XML; displaying source. <p>Since Mike's thread <a href="">questions on structured (oo,1)-topos</a> got a bit highjacked by general oo-stack questions, I thought I'd start this new thread to announce attempted answers:</p> <p>Mike had rightly complained in a query box that a "remark" of mine in which I had meant to indicate the intuitive meaning of the technical condition on an (oo,1)-structure sheaf had been "ridiculous".</p> <p>But that condition is important, and important to understand. I have now removed the nonsensical paragraph and Mike's query box complaining about it, inserted a new query box saying "second attempt" and then spelled out two archetypical toy examples in detail, that illustrate what's going on.</p> <p>The second of them can be found in StrucSp itself, as indicated. It serves mainly to show that an ordinary ringed space has a structure sheaf in the sense of structured oo-toposes precisely if it is a <a href="">locally ringed space</a>.</p> <p>But to try to bring out the very simple geometry behind this even better, I preceded this example now by one where a structure sheaf just of continuous functions is considered.</p> <p>Have a look.</p>
    • can anyone point me to some useful discussion of cosismplicial simplicial abelian groups

       \Delta \to [\Delta^{op}, Ab]

      and cosimplicial simplicial rings

       \Delta \to [\Delta^{op}, CRing]

      I guess there should be a Dold-Kan correspondence relating these to unbounded (co)chain complexes (that may be nontrivial both in positive as well as in negative degree). I suppose it's kind of straightforward how this should work, but I'd still ike to know of any literature that might discuss this. Anything?

    • I expanded the section Gradings at cohomology. Made three sub-paragraphs:

      • integer grading

      • bigrading

      • exoctic grading

      using the kind of insights that we were discussing recently in various places.

      I also put in a query box where I wonder about a nice way to define a Chern character construction for general oo-stack oo-toposes.

    • added an interesting reference by Kriegl and Michor to generalized smooth algebra, kindly pointed out by Thomas Nikolaus:

      C^\infty-algebras from the functional analyytic point of view

      Also added some other references.

    • I completely rewrote BrownAHT

      same message as before, but I think I could say it much better now