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    • At synthetic differential infinity-groupoid I have entered statement and detailed proof that flat and infinitesimally flat real coefficients are equivalent in SynthDiff∞GrpdSynthDiff\infty Grpd

      ♭ infB nℝ≃♭B nℝ. \mathbf{\flat}_{inf} \mathbf{B}^n \mathbb{R} \simeq \mathbf{\flat} \mathbf{B}^n \mathbb{R} \,.

      The proof proceeds by presentation of ♭ infB nℝ\mathbf{\flat}_{inf} \mathbf{B}^n \mathbb{R} by essentially (a cofibrant resolution of) Anders Kocks’ s infinitesimal singular simplicial complex. In this presentation cohomology with coefficients in this object is manifestly computed as in de Rham space/Grothendieck descent-technology for oo-stacks.

      But we also have an intrinsic notion of de Rham cohomology in cohesive ∞\infty-toposes, and the above implies that in degree n≥2n \geq 2 this coincides with the de Rham space presentation as well as the intrinsic real cohomoloy.

      All in all, this proves what Simpson-Teleman called the “de Rham theorem for ∞\infty-stacks” in a note that is linked in the above entry. They consider a slightly different site of which I don’t know if it is cohesive, but apart from that their model category theoretic setup is pretty much exactly that which goes into the above proof. They don’t actually give a proof in this unpublished and sketchy note and they work (or at least speak) only in homotopy categories. But it’s all “morally the same”. For some value of “morally”.

    • A manifold has

      • a set of orientations;

      • an xyz of topological spin structures

      • a 3-groupoid of topological string structures;

      • a 7-groupoid of topological fivebrane stuctures, etc.

      and for some reason it is common in the literature (which of course is small in the last cases) to speak of these nn-groupoids, but not so common to speak of the xyz here:

      • A manifold has a groupoid of spin structures.

      Namely the homotopy fiber of the second Stiefel-Whitney class

      Spin(X)→Top(X,BSO)→(w 2) *Top(X,B 2ℤ 2). Spin(X) \to Top(X,B SO) \stackrel{(w_2)_*}{\to} Top(X, B^2 \mathbb{Z}_2) \,.

      I have added one reference that explicitly discusses the groupoid of spin structures to spin structure.

      Do you have further references?

    • I had created line Lie n-algebra, just for the sake of completeness and so that I know where to link to when I mention it

    • I have created an entry differential characteristic class.

      I felt need for this as the traditional term secondary characteristic class first of all has (as discussed there) quite a bit of variance in convention of meaning in the established literature, and secondly it is unfortunately undescriptive (which is probably the reason for the first problem, I guess!).

      Moreover, I felt the need for a place to discuss the concept “differential characteristic class” in the fully general abstract way in the spirit of our entry on cohomology, whereas “secondary characteristic class” has a certain association with concrete models. Some people use it almost synonymously with “Cheeger-Simons differential character”.

      Anyway, so I created a new entry. So far it contains just the “unrefined” definition. I’ll try to expand on it later,

    • I notice that the entry essential image is in a bad state:

      it starts out making two statements, the first of which is then doubted by Mike in a query box, the second doubted by Zoran in a query box.

      If there is really no agreement on what should go there, we should maybe better clear the entry, and discuss the matter here until we have a minimum of consensus.

      But I guess the problems can easily be dealt with and somebody should try to polish this entry right away.

    • I have taken this opportunity to update the references section at profunctor, based on recent emails from Marta Bunge and Jean Benabou.

      I have added a little detail to the comment at anafunctor that Kelly considered anafunctors without naming them, namely the paper and the year, and also a small concession to Jean Benabou who wanted it widely known that he recently discovered the equivalence between anafunctors and representable profunctors viz, naming him explicitly at the appropriate point of the discussion.

      (I do not want to drag the recent discussion held on and off the categories mailing list here - I just wanted to make the changes public)

    • quick stub for volume form, as I need the link somewhere for completeness

    • I’ve decided that these shouldn’t exist (making me agree with the standard terminology) and explained why at regular cardinal.

    • Do we have a discussion anywhere that 2-limits in the (2,1)-category of categories as defined in the 2-category-literature do coincide with the coresponding limits computed inside the (∞,1)(\infty,1)-category of (∞,1)(\infty,1)-categories?

      I thought we had, but maybe we don’t. If not, I’ll try to add some discussion.

    • I split off (2,1)-algebraic theory of E-infinity algebras, but it’s still the same stubby context as before.

      (I will probably/hopefully fill in more details in two weeks, as preparation for one of the sessions of our derived geometry semninar)

    • we are lacking content in the entry topos theory.

      I added a one-line Idea and then expanded the list of references.

    • Added to 2-monad a remark about Power’s result that any monad on the underlying category of a strict 2-category with powers or copowers has at most one enrichment to a strict 2-monad.

    • Have created an entry TopMfd

      (this is supposed to be in the tradition that with the entry topological manifold that discusses the properties of the objects we also have an entry that discusses the properties of the category that these objects form).

    • expanded and polished the entry model structure on simplicial sheaves (to be distinguished from the one of simplicial pre-sheaves!)

      Made explicit the little corollary that for D→CD \to C a dense sub-site, the corresponding hypercompleted ∞\infty-sheaf ∞\infty-toposes are equivalent.

    • there is some confusion on this MO thread about sheafification, with the nnLab entry sheafification somehow involved. I had a look at the entry and find that it can do with lots of polishing, but that the statement discussed over there is clearly right. (the misleading answer on MO that seems to claim a problem on the nLab page gets twice as many votes as the good answer by Clark Barwick, which confirms the statement) I have tried to edit it a bit to make things clearer, but don’t have the leisure for that now.

      Given the recent success with the polishing of the entry on geometric realization, maybe I should announce that sheafification is going to be submitted for nnJournal peer-review soon, so that everybody here will jump on it to brush it up ;-)

    • Created pro-set with an adjunction and a counterexample.

    • Behind the scenes Domenico Fiorenza is having a long discussion with me and Jim Stasheff on the matters that are being discussed at differential cohomology in an (oo,1)-topos – examples. It seems we want to work on this together. Accordingly, I have now moved at least parts of this to the main nLab in the new entry

      infinity-Chern-Weil theory.

      I added a remark right at the beginning that is supposed to indicate the nature of this material.

    • Hello. I’ve taken up a new cause. I made an article about schlessinger’s criterion. There seems to be very little about the higher category perspective on deformation theory. This is what I’m really interested in as a grad student, so I thought I’d try to fill in a few holes.

    • I rewrote a good bit of the entry sheaf, trying to polish and strengthen the exposition.

      The rewritten material is what now constituttes the section “Definition”. This subsumes essentially everything that was there before, except for some scattered remarks which I removed and instad provided hyperlinks for, since they have meanwhile better discussions in other entries.

      I left the discussion of sheaves and the general notion of localization untouched (it is now in the section “Sheaves” and localization”). This would now need to be harmonized notationally a bit better. Maybe later.

    • Had a problem with guest posting, just testing ... nothing to see here ...

      (Andrew Stacey)

    • I’ve added a section called 𝒜−gerbes\mathcal{A}-gerbes at gerbe (as a stack) in an attempt to add something about the differential geometry question that was raised. I’m just a lowly grad student so be gentle if I’ve accidentally written something crazy.

    • We don’t use the term “range” much here, and I explained why.

    • added to filtered (infinity,1)-category the statement that these are precisely those shapes of diagrams such that ∞\infty-colimits over them commute with finite ∞\infty-limits.

    • algebrad and additions at Nikolai Durov. The movie starts slow and boring but gets very interesting after a while when the topic develops.

    • I have edited posite

      In particular I tried to work the query box into the text. Mike and David R. please check if you agree with the result.

    • New entry thread needed at newly expanded pro-object. Adequate changes at filtered limit.

      While threads are elements in the cofiltered limit of sets or spaces, germs are classes of equivalence which appear in the treatment of filtered colimits. However the entry germ is taking germs just in the special case of colimits forming stalks of an etale space. In my practice the notion of germ could be used more generally for directed colimits or even filtered colimits in groups, sets, and alike. Is it only me using the terminology in this extended sense ? I would like to know the opinions.

    • added to infinity-cohesive site statement and proof that if all objects of the ∞\infty-cohesive site have points, then the cohesive ∞\infty-topos over it satisfies the axiomm pieces have points .

      (Easy proof using the previous results and Dugger’s cofibrant replacement theorem for [C op,sSet] proj,loc[C^{op}, sSet]_{proj,loc}).

    • added to over-(infinity,1)-category the statement that the hom-spaces are computes by homotopy fibers of hom-spaces in the underlying ∞\infty-category in direct analogy to the 1-categorical case

    • added at local (infinity,1)-topos

      • statement and proof that over ∞Grpd\infty Grpd the ∞\infty-connectivity condition is redundant;

      • statement and proof that for any ∞\infty-topos H\mathbf{H} over ∞Grpd\infty Grpd and X∈HX \in \mathbf{H} any small-projective object, the slice H/X\mathbf{H}/X is local.

    • because I needed the link I have created a stub (2,1)-site. I define it there as an (infinity,1)-site whose underlying category is a (2,1)-category. Don’t have the leisure right now to check that this is also a 2-site whose underlying 2-category is a (2,1)(2,1)-category, as hopefully it is

    • have created infinity-connected site

      • moved over from infinity-cohesive site the proof that the ∞\infty-topos over a locally and globally ∞\infty-connected site is locally and globally ∞\infty-connected; (and had occasion to polish and streamline it a bit more)

    • created strongly infinity-connected site with the analogous definition, analogous proposition and analogous proof as at strongly connected site.

      Only difference is that I define it as a cosifted ∞\infty-connected site instead of as a cosifted locally ∞\infty-connected site, because I am currently not quite sure about the definition of the latter because my model-category theoretic proofs rely on the existence of a terminal object, without that my standard Quillen adjunction model for the terminal ∞\infty-geometric morphism fails and I’d need to think harder.

      But all these entries of higher connected and local sites currently have the issue that they give sufficient conditions and don’t prove necessary conditions, so I think it’s okay, but we should keep in mind that there might be refinements of these definitions.

    • created connected site and declared it to be a locally connected site with a terminal object.

      That’s sufficient for its sheaf topos to be connected, I don’t now if it is necessary: If we find a weaker sufficient condition we should refine the entry.

    • I plan to write few foundations/set theory stubs including Skolem paradox. It will wait for a bit as the nnLab seems to be down at the moment.

      The entry forcing has phrase downward Löwenheim-Skolem theorem. What does it mean downward in this phrase ? Is it a modifier at all ?

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