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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• have created an entry Khovanov homology, so far containing only some references and a little paragraph on the recent advances in identifying the corresponding TQFT. I have also posted this to the $n$Café here, hoping that others feel inspired to work on expanding this entry

• I wanted to understand Milnor’s paper on Link Groups, so I basically rewrote the main bits in to Milnor mu-bar invariants. (I don’t understand the difference between $\mu$-invariants and $\bar{\mu}$-invariants, but I was only working on the original paper so presumably haven’t gotten that far yet.)

I even put a TOC in so Urs will be happy!

• I have just added a link to the notes that I prepared for the Lisbon meeting on my personal page. I would love to have some feedback, and in particular suggestions for incorporating some more of this in the nLab. The new material also forms part of the extended version of the Menagerie (which is now topping 800 pages.)

• At string 2-group it is claimed that the sequence of classifying spaces ending --> BSO(n) --> BO(n) is the Whitehead tower of O(n). Also mentioned is the version for smooth infinity groupoids (so I assume it is Urs who put that there). It is certainly not true that the sequence of classifying spaces so stated is the Whitehead tower for O(n), but the details for groups considered as one-object infinity groupoids are open to interpretation, so I haven't changed anything. Just a heads up.

-David Roberts
• I have added a brief note about type-theoretic polymorphism to the list of impredicative axioms at predicative mathematics.

• Added the statement of the Isbell-Freyd characterization of concrete categories, in the special case of finitely complete categories for which it looks more familiar, along with the proof of necessity.

• At effects of foundations on “real” mathematics I’ve put in the example of Fermat’s last theorem as being potentially derivable from PA, and pointed to two articles by McLarty on this topic.

(Edit: the naive wikilink to the given page breaks, due to the ” ” pair)

• You may or may not recall the observation, recorded at Lie group cohomology, that there is a natural map from the Segal-Blanc-Brylinski refinement of Lie group cohomology to the intrinsic cohomology of Lie groups when regarded as smooth infinity-groupoids.

For a while i did not know how to see whether this natural map is an equivalence, as one would hope it is. The subtlety is that the Cech-formula that Brylinski gives for refined Lie group cohomology corresponds to making a degreewise cofibrant replacement of $\mathbf{B}G$ in $Smooth \infty Grpd$ and then taking the diagonal, and there is no reason that this diagonal is itself still cofibrant (and I don’t think it is). So while Segal-Brylinski Lie group cohomology is finer and less naive than naive Lie group cohomology, it wasn’t clear (to me) that it is fine enough and reproduces the “correct” cohomology.

So one had to argue that for certain coefficients the degreewise cofibrant resolution in $[CartSp^{op}, sSet]_{proj,loc}$ is already sufficient for computing the derived hom space. It was only yesterday that I realized that this is a corollary of the general result at function algebras on infinity-stacks once we embed smooth infinity-groupoid into synthetic differential infinity-groupoids.

So I believe I have a proof now. I have written it out in synthetic differential infinity-groupoid in the section Cohomology and principal $\infty$-bundles.

• 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 hadd added a little bit of this and that to category of cobordisms earlier today in a prolonged coffee break.

This was in reaction to learning about the work by Ayala, now referenced there, whou considers categories of cobordisms equipped with geometric structure given by morphisms into an $\infty$-stack $\mathcal{F}$.

• A while back I had a discussion here with Domenico on how the framed cobordim $(\infty,n)$-category $Bord^{fr}_n(X)$ of cobordisns in a topological space $X$ should be essentially the free symmetric monoidal $(\infty,n)$-category on the fundamental $\infty$-groupoid of $X$.

This can be read as saying

Every flat $\infty$-parallel transport of fully dualizable objects has a unique $\infty$-holonomy.

(!)

Some helpful discussion with Chris Schommer-Pries tonight revealed that this is (unsurprisingly) already a special case of what Jacob Lurie proves. He proves it in more generality, which makes the statement easy to miss on casual reading. So I made it explicit now at cobordism hypothesis in the new section For cobordisms in a manifold.

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

$\mathbf{\flat}_{inf} \mathbf{B}^n \mathbb{R} \simeq \mathbf{\flat} \mathbf{B}^n \mathbb{R} \,.$

The proof proceeds by presentation of $\mathbf{\flat}_{inf} \mathbf{B}^n \mathbb{R}$ 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 $n \geq 2$ 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 $n$-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

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

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)

• I have renamed the entry formerly called (and still redirecting) “connection on a principal infinity-bundle” into connection on a smooth principal infinity-bundle.

In the same vein I have renamed the entry formerly titled (and still redirecting) “infinity-Chern-Weil theory” into Chern-Weil theory in Smooth∞Grpd.

This way things are set up well for when the legions of students arrive who will do all the analogous discussion in other cohesive $(\infty,1)$-toposes such as $Algebraic \infty Grpd$, $ComplexAnalytic \infty Grpd$ as well as the derived version of all of these. ;-)

• 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 $(\infty,1)$-category of $(\infty,1)$-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 $D \to C$ 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 $n$Lab 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 $n$Journal 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

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 $\mathcal{A}-gerbes$ 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.