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

• 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.

• 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.

• this is a message to Zoran:

I have tried to refine the section-outline at localizing subcategory a bit. Can you live with the result? Let me know.

• discovered the following remnant discussion at full functor, which hereby I move from there to here

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.

Mike says: Do you have a reason to care about full functors which are not also faithful? I’ve never seen a very compelling one. (Maybe I should just read your thesis…) I agree that “full morphism” (in the representable sense) is not really a useful/correct concept in a general 2-category, and that therefore “full and faithful” is not entirely appropriate, so I usually use “ff” in that context. I’ve changed the entry above a bit to reflect your comment; is it satisfactory now? Maybe all this should actually go at full and faithful functor (and/or fully faithful functor)?

• I am starting stubs

• created an entry mapping cocone, following a suggestion by Zoran, that this is the right technical term for what is discussed in more detail at generalized universal bundle.

(the examples section needs more attention, though...)

• I have created final lift, and added to adjoint triple a proof that in a fully faithful adjoint triple between cocomplete categories, the middle functor admits final lifts of small structured sinks (and dually). This means that it is kind of like a topological concrete category, except that the forgetful functor need not be faithful.

I find this interesting because it means that in the situation of axiomatic cohesion, where the forgetful functor from “spaces” is not necessarily faithful, we can still construct such “spaces” in “initial” and “final” ways, as long as we restrict to small sources and sinks.

• If you're not following the categories mailing list, then you're missing out on a great discussion of evil. Peter Selinger has come from the list to the Lab to discuss it here too!

• Thought I’d write up some old notes at symmetric product of circles (linked from unitary group, explanation to come on symmetric product of circles). Not finished yet, but have to leave it for now.

(I was incensed to discover that to look at the source article for the material for this to check that I’m remembering it right - I last looked at it about 10 years ago - I have to pay 30 UKP. The article is 3 pages long. That’s 10UKP per page! So I’m going from vague memories and “working it out afresh”.)

• Added Thom-Federer and Gottlieb thorems to Eilenberg-MacLane space; added the remark “$\Omega\mathbf{C}(X,Y)\simeq \mathbf{C}(X,\Omega Y)$ in any (oo,1)-category with homotopy pullbacks” in loop space object.

• Partially spurred on by an MO question, I have started an entry on simple homotopy theory. I am also intrigued as to whether there is a constructive simple homotopy theory that may apply in homotopy type theory, but know so little (as yet) about that subject that this may be far fetched.