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

• 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?
• 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?
• I have created a stub entry for A. Suslin. Can someone add in the Russian original of his name please, as I do not know if the Wikipedia version is correct?

• Some time ago, I split Cheng space from measurable space, but I never announced it here (nor removed if from the list of things to do at the latter). Note: Henry Cheng, not Eugenia Cheng.

• I added a brief equivalence between two notions of characters of profinite groups that I spotted on MO.

• I have created an entry Wu class.

At the end I have also included an “Applications”-section with comments on Wu classes in the definition of higher Chern-Simons functionals. That eventually deserves to go in a dedicated entry, but for the moment I think it is good to have it there, as it is a major source of discussion of Wu structures in the maths literature, quite indepently of its role in physics.

• I added links to the horizontal categorifications in group object and created groupoid object.

In groupoid object in an (infinity,1)-category I read the conspicious statement: ”an internal ∞-group or internal ∞-groupoid may be defined as a group(oid) object internal to an (∞,1)-category C with pullbacks” - but this terminology seems to hinder distinguishing between them and ∞-groupoid objects in (∞,1)-categories.

• After Mike’s post, scone was created. But I see at Freyd cover it says

The Freyd cover of a category – sometimes known as the Sierpinski cone or “scone” – is a special case of Artin gluing

Are they synonyms?

• I am just hearing about the Alfsen-Shultz theorem about states on C*-algebras, so I started a stub to remind me. Still need to track down the reference and the details.

• I went through locale and made some of the language consistent throughout the article. Also I added a new section, Subsidiary notions, to which I intend to add.

• started a stub entry Toposes on the category (2-category) of all toposes. But nothing much there yet.

• when creating a stub entry local Langlands conjectures I noticed that it has already become hard to know which entries on the Langlands program we already have. I always take this as a sign that a summary table of contents is called for. So I started

Langlands correspondence – contents

(Even though all of these entries are still more or less stubs.)

• I noticed that there was some wild formatting at building. I have tried to tame it a bit.

• I have started a table of contents measure theory - contents and started adding it as a floating toc to the relevant entries

• currently the bulk of the entry analytic geometry is occupied by a long section on “Holomorphic functions of several complex variables”. Should that not better be moved to some dedicated entry of its own? Any opinions?

• I created branched manifold -linked from orbifold- with a definition from ”expanding attractors” by Robert F. Williams (1974) quoted in wikipedia. This description is -as it stands- not precisely compatible to that given in Dusa McDuffs ”Groupoids, Branched Manifolds and Multisections” which I am rather interested in. So I plan to comment on this as a side note in the -yet to be written-article orbifold groupoid.

• New article: cofinality, with the basic case being the cofinality of a quasiorder as a collection of cardinal numbers, a variation as a collection of ordinal numbers (or equivalently an ordinal number), and an apparently separate case of the cofinality of a collection of cardinal numbers, all of these tied together and interpreted as a single cardinal number if one assumes the axiom of choice.

• added to diffeomorphism group statements and references for the case of 3-manifolds (Smale conjecture etc.)

• I was surprised to discover that we had no page finite (infinity,1)-limit yet, especially given that they are slightly subtle in relation to the 1-categorical version. So I made one.

• Theorem: The existence of arbitrarily large supercompact cardinals implies the statement:

Every absolute epireflective class of objects in a balanced accessible category is a small-orthogonality class.

In other words, if $L$ is a reflective localization functor on a balanced accessible category such that the unit morphism $X \to L X$ is an epimorphism for all $X$ and the class of $L$-local objects is defined by an absolute formula, then the existence of a suficciently large supercompact cardinal implies that $L$ is a localization with respect to some set of morphisms.

This is in BagariaCasacubertaMathias

Urs Schreiber: I am being told in prvivate communication that the assumption of epis can actually be dropped. A refined result is due out soon.

Mathieu says: I agree that, for functors, there is no reason to say “fully faithful” rather than “full and faithful”. But for arrows in a 2-category (like in the new version of the entry on subcategories), there are reasons. I quote myself (from my thesis): «Remark: we say fully faithful and not full and faithful, because the condition that, for all $X:\C$, $C(X,f)$ be full is not equivalent in $\Grpd$ to $f$ being full. Moreover, in $\Grpd$, this condition implies faithfulness. We will define (Definition 197) a notion of full arrow in a $\Grpd$-category which, in $\Grpd$ and $\Symm2\Grp$ (symmetric 2-groups), gives back the ordinary full functors.» Note that this works only for some good groupoid enriched categories, not for $\Cat$, for example.