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• I have started an entry canonical extension.

• I am fiddling with an entry table - models for (infinity,1)-operads meant to allow to see 10+ different model categories and their main Quillen equivalences at one glance.

I guess there are better ways to typeset this. (Volunteers please feel free to lend a hand!) But for the time being I’ll settle with what I have so far.

• Added a section arithmetic D-modules. This is the optimal theory for p-adic cohomology of varieties over finite fields, since it has the six operations. This section is complementary to rigid cohomology.

• In

http://ncatlab.org/nlab/show/Lie+2-algebra, at

“… the differential respects the brackets: for all $x \in g_0$ and $h \in g_1$ we have

$\delta [x,h]=[x,\delta h]$

…”

is wrong. The equation should be:

$\delta \alpha(x,h) = [x,\delta h]$

Since I don’t know if I have the right to change an nLab entry,I post this here as an suggestion.

• Created a new entry on "special $\Delta$-spaces". Does anyone know of a better name?

The entry is poorly edited since I'm not fluent with the iTex syntax.
• I made some much-needed corrections at simplicial complex, directed mostly at errors which had been introduced by yours truly. I also created quasi-topological space (the notion due to Spanier).

I haven’t thought this through, but regarding the process of turning a simplicial complex into a simplicial set, the usual sequence of words seems to involve putting a non-canonical ordering on the set of vertices and then getting ordered simplices from that. But is there anything “wrong” with taking the composite

$SimpComp \hookrightarrow Set^{Fin_{+}^{op}} \stackrel{Set^{i^{op}}}{\to} Set^{\Delta^{op}}$

where the inclusion is the realization of simplicial complexes as concrete presheaves on nonempty finite sets, and the second arrow is pulling back along the forgetful functor $i$ from nonempty totally ordered finite sets to nonempty finite sets? This looks much more canonical.

• I hope Urs doesn’t mind my inserting a not-too-serious but nevertheless amusing example at symplectomorphism.

• Someone set up lawvere theory, but did not add anything to it. They had previously done an edit to FinSet. The new page has a redirect from Lawvere+theory, so I don’t see what 88.104.160.245 is doing. Can someone check the edit at [[FinSet]. It looks as if the person knows some things and so has added a bit, but it is so long since I knew that stuff well so I cannot tell if it is a valid edit or not.

• I have created a stub on Volodin. I have been unable to find out more on him. Can anyone help?

• Started presentation of a category by generators and relations. This is probably an evil definition (there was an old discussion on this in the context of quotient category), and there is perhaps a more modern way to do this, so feel free to change the entry. I used “quotient category” as in CWM and mentioned that this is not the definition in the nLab.

• I have added some material into Vietoris complex relating to links with algebraic K-theory and some conjectures of Kapranov and Saito.

• The last two days Stephan Spahn was visiting me, and we chatted a lot about étaleness in cohesive ∞-toposes.

We found proofs that

• for every notion of infinitesimal cohesive neighbourhood

$i : \mathbf{H} \hookrightarrow \mathbf{H}_{th}$

the total space projections of locally constant $\infty$-stacks are formally étale;

• the formally étale morphisms with respect to any choice of infinitesimal cohesion satisfy all the axioms of axiomatic open maps (or rather their $\infty$-version, of course).

(These are to be written up. Requires plenty of 3d iterated $\infty$-pullback diagrams which are hard to typeset).

Recall – from synthetic differential infinity-groupoid – that for the infinitesimal cohesive neighbourhood

$i :$ Smooth∞Grpd $\hookrightarrow$ SynthDiff∞Grpd

the axiomatically formally étale morphisms between smooth manifolds are precisely the étale maps in the traditional sense.

Motivated by all this, I finally see, I think, what the correct definition of cohesive étale ∞-groupoid is:

simply: $X \in \mathbf{H}$ is an étale cohesive $\infty$-groupoid if it admits an atlas $X_0 \to X$ by a formally étale morphism in $\mathbf{H}$.

I have spelled out the proof now here that with this definition a Lie groupoid $\mathcal{G}$ is an étale groupoid in the traditional sense, precisely if it is cohesively étale when regarded as an object of the infinitesimal cohesive neighbourhood $Smooth \infty Grpd \hookrightarrow SynthDiff \infty Grpd$.

I hope to further expand on all this with Stephan. But I may be absorbed with other things. Next week I am in Goettingen, busy with a seminar on $L_\infty$-connections.

• I found in the disambiguation page basis an entry basis of a vector space missing, and so I created one. I have then cross-linked this with relevant existing entries, such as basis theorem. Also created a brief entry orthogonal basis in order to get rid of grayish links is some entries.

• In context of size issues on admissible structures (in the sense of DAG V) I wondered which closure properties (e.g. obviously its closed under pullbacks along relatively k-compact morphisms) the class of relatively k-compact (for a regular cardinal k) morphisms in a (∞,1)-category satisfies. Is there any reference concerning this?

• I needed an entry that lists references on twisted K-homology, so I created one. This made me notice that we currently lack an entry K-homology. I can try to create a stub for that a little later…

• the link to the picture in the entry Charles Wells is broken. Does anyone know how to fix it or have an alternative picture?

• Perhaps we need a page on Jean-Louis Koszul, and possibly an agreement on terminology / names for entries. Ben has created a page called Koszul, but usually single names like that would be used for the ’person’ page of that mathematician, so … The term in in any case (as adjective) is also used in various other contexts e.g. for operads, so possibly there needs to be some rationalisation. My thought would be to combine a page on Koszul (and the Wikipedia (English) page on him is poor, and includes some very poor translation from the French ) with a certain amount of disambiguation, however I am not an expert on things ’Koszul’ and this may not be the most efficient way to go.

• I created orbifold groupoids with some classes of groupoids whose elements I like to think of as ‘‘orbifold groupoids‘‘. It would be nice to have a discussion of the interelation of these classes there, too.

• I was in the process of creating an entry for Cartesian fibration of dendroidal sets, when by accident I suddently discovered that the degree-1 case of this had been considered before, by Claudio Hermida. So now I have also created a brief entry

• I thought it would be useful to supplement the entry operad with entries symmetric operad and planar operad, that amplify a bit more on the specifics of these respective flavors of the general notion, and that will allow us in other entries to link specifically to one of the two notions, when the choice is to be made explicit.

So far I have written (only) an Idea-section at symmetric operad with some comments.

To be expanded.

• I have created a bunch of stub entries such as iterated loop space object with little non-redundant content for the moment. I am filling the k-monoidal table. Please bear with me for the time being, while I add stuff.

• have created an entry k-monoidal table to be used for inclusion into the entries that it organizes (see for instance at infinite loop space).

Will now create at least stubs for the missing links.

• I have created an entry on the Steinberg group $St(R)$ of a ring $R$. The entry includes the Whitehead lemma.

• I have half-heartedly started adding something to Kac-Moody algebra. Mostly refrences so far. But I don’t have the time right now to do any more.

• Is the notion of local Kan extension in weak or infinity n-categories well defined. I know there is the infinity,1-case done by Lurie, but it is not local. I would define a Kan extension as the datum of a 1-morphism filling the usual diagram, with a 2-morphism that induces an equivalence of (n-2)-categories of morphisms between morphisms. You may do the same in the infinity,n case, if needed.

Is there a better definition?

The point is to define limits in weak n-categories using this. It is not simpler to define limit than to define Kan extensions.

Are there finer notions of local extensions, that use more explicitely the higher category structure?
• I have started a stub on this. (Isn’t there a nice nPOV interpretation of this?)

• A new contribution
• There is a strange glitch on this page: the geometric realization of a cubical set (see geometric realizationealization) below) tends to have the wrong homotopy type:

That is what appears but is not a t all what the source looks like:

the geometric realization of a cubical set (see [geometric realization](#geometric realization) below) tends to have the wrong homotopy type:

What is going wrong and how can it be fixed?

Another point : does anyone know anything about symmetric cubical sets?

• I made some edits at classifying topos to correct what I thought were some inaccuracies. One is that simplicial sets classify interval objects, but offhand I didn’t see the exact notion of linear interval over at interval object that would make this a correct sentence. In any event, I went ahead and defined the notion of linear interval as a model of a specified geometric theory.

The other is for local rings. I think when algebraic geometers refer to a sheaf of local rings, they refer to a sheaf of rings over a (sober) space all of whose stalks are local. I wasn’t sure that description would be kosher for a general (Grothendieck) topos $E$ since there may not be any “stalks” (i.e., points $Set \to E$) to refer to. In any case, it seems to me safer to give the geometric theory directly.

• Matan Prezma kindly pointed out the he has an article with a correction to what used to be prop. 2 at model structure on cosimplicial simplicial sets. (One has to use restricted totalization instead of ordinary totalization.)

I have corrected this and added the reference.

I also added to the entry a remark that makes the relation to descent objects explicit. Right now this is remark 2.

• at subgroup I have collected a list of links to $n$Lab entries of the form “xyz subgroup”. If anyone knows of more such entries (or does a more thorough search), please consider adding them!

• How is Majid's bialgebra cohomology related to GS cohomology for bialgebras?

I saw the recent paper by Shoikhet where GS cohomology is k-monoidally understood.

Is there a similar understanding of Majid's cohomology?