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- Discussion Type
- discussion topicsemi-simplicial set
- Category Latest Changes
- Started by Mike Shulman
- Comments 22
- Last comment by Thomas Holder
- Last Active Apr 17th 2023

Created semi-simplicial set, mainly as a repository for some terminological remarks. I would welcome anyone more knowledgeable about the history to correct or improve it!

- Discussion Type
- discussion topicstringor bundle
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Apr 17th 2023

- Discussion Type
- discussion topiczero bundle
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 17th 2023

a fairly trivial entry, for the purpose of easing the hyperlinking in entries like

*bireflective subcategory*,*parameterized spectra*,*VectBund*, etc.

- Discussion Type
- discussion topicexcisive (∞,1)-functor
- Category Latest Changes
- Started by David_Corfield
- Comments 2
- Last comment by Urs
- Last Active Apr 17th 2023

Added the reference now that it has appeared:

- {#AnelBiedermanFinsterJoyal17b} Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal,
*Goodwillie’s Calculus of Functors and Higher Topos Theory*(arXiv:1703.09632)

- {#AnelBiedermanFinsterJoyal17b} Mathieu Anel, Georg Biedermann, Eric Finster, André Joyal,

- Discussion Type
- discussion topicJeroen Hekking
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 17th 2023

- Discussion Type
- discussion topicderived blow-up
- Category Latest Changes
- Started by David_Corfield
- Comments 2
- Last comment by Urs
- Last Active Apr 17th 2023

- Discussion Type
- discussion topicHans-Werner Henn
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 17th 2023

- Discussion Type
- discussion topicgroup cohomology
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active Apr 17th 2023

added to group cohomology

in the section structured group cohomology some remarks about how to correctly define Lie group cohomology and topological group cohomology etc. and how not to

in the section Lie group cohiomology a derivation of how from the right oo-categorical definition one finds after some unwinding the correct definition as given in the article by Brylinski cited there.

it's late here and I am now in a bit of a hurry to call it quits, so the proof I give there may need a bit polishing. I'll take care of that later...

- Discussion Type
- discussion topiclocalization
- Category Latest Changes
- Started by John Baez
- Comments 30
- Last comment by Urs
- Last Active Apr 17th 2023

- I added some basic explanatory stuff to the "Idea" section of localization. The simple idea of localizing a commutative ring really does, I think, correspond to some fancier operation you can do in topos theory (namely the topos of sheaves on the spectrum of that ring?). So, someone should try to explain that... but I can't.

Someone should also add links from "localization" to various fancier pages on fancier kinds of localization. I did so for Bousfield localization. I also added a bit more explanation to that page.

(I think my job at the nLab is to make pages easier for beginners to follow. This is a gargantuan, almost hopeless task... but I don't see anything else I'm dying to do here. So I'll just do a little. It makes sense for me to add more explanation to an nLab page whenever I plan to cite it on This Week's Finds.)

- Discussion Type
- discussion topicmodel structure for excisive functors
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 2
- Last comment by Urs
- Last Active Apr 17th 2023

- Discussion Type
- discussion topicdecategorification
- Category Latest Changes
- Started by bwebster
- Comments 14
- Last comment by Urs
- Last Active Apr 17th 2023

I thought it might be good if somebody explained the relationship between decategorification and extended TQFT. My understanding from talking to physicists is that you should multiply your space by $S^1$; is this right in a mathematical sense? I've added a query box asking roughly the same thing.

Also, I attempted to add a sidebar, mostly just to try it out, and somehow it's not rendering right. Anyone want to explain what I did wrong?

- Discussion Type
- discussion topicMagnus Forrester-Barker
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 16th 2023

- Discussion Type
- discussion topicinternalization
- Category Latest Changes
- Started by Urs
- Comments 44
- Last comment by Urs
- Last Active Apr 16th 2023

started to add to internalization a list of links to examples. Probably we have much more.

- Discussion Type
- discussion topicdescent
- Category Latest Changes
- Started by Tim_Porter
- Comments 4
- Last comment by Urs
- Last Active Apr 16th 2023

I made some very minor changes to the introduction at descent. I hesitate to do more but at present the discussion does not seem that readable to me. Can someone look at it to see what they think? The intro seems to plunge in deep very quickly and so the ‘idea’ of descent as that of gluing local information together, does not come across to me. The article is lso quite long and perhaps needs splitting up a bit.

- Discussion Type
- discussion topicdescent morphism
- Category Latest Changes
- Started by Mike Shulman
- Comments 28
- Last comment by Urs
- Last Active Apr 16th 2023

Created descent morphism.

In adding links, I discovered that Euclidean-topological infinity-groupoid and separated (infinity,1)-presheaf use the phrase “descent morphism” to refer to the

*comparison functor*mapping into the category of descent data. If no one has any objections, I would like to change this to avoid confusion, but I’m not sure what to change it to: would “comparison functor” be good enough?

- Discussion Type
- discussion topicquotient stack
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by Urs
- Last Active Apr 16th 2023

I noticed that there was no entry

*quotient stack*, so I quickly started one, just to be able to point to it from elswhere.

- Discussion Type
- discussion topicalgebraic stack
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Urs
- Last Active Apr 16th 2023

brushed up the definition at algebraic stack following the

*Stacks project*text..

- Discussion Type
- discussion topicArtin stack
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Apr 16th 2023

finally created stub for Artin stack

also rewrote and expanded the Idea-section at Deligne-Mumford stack

a few days back I had worked my old query box at algebraic stack into the definition. But hastily so.

All three of these entries (algebraic stack, Deligne-Mumford stack and Artin stack ) need careful polishing and expansion. My suggestion is that we eventually expand algebraic stack to a comprehensive discussion and have the other two be more or less clarification of terminology and otherwise be just redirects.

- Discussion Type
- discussion topicDaniel Halpern-Leistner
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 16th 2023

- Discussion Type
- discussion topicMichael Groechenig
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 16th 2023

- Discussion Type
- discussion topiccanonical topology
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Apr 16th 2023

somebody asked me for the proof of the claim at

*canonical topology*that for a Grothendieck topos $\mathbf{H}$ we have $\mathbf{H} \simeq Sh_{can}(\mathbf{H})$.I have added to the entry pointers to the proof in Johnstone’s book, and to related discussion for $\infty$-toposes. Myself I don’t have more time right now, but maybe somebody feels inspired to write out some details in the nLab entry itself?

- Discussion Type
- discussion topicHomePage
- Category Latest Changes
- Started by Urs
- Comments 82
- Last comment by Urs
- Last Active Apr 16th 2023

I have added at

*HomePage*in the section*Discussion*a new sentence with a new link:If you do contribute to the nLab, you are strongly encouraged to similarly drop a short note there about what you have done – or maybe just about what you plan to do or even what you would like others to do. See Welcome to the nForum (nlabmeta) for more information.

I had completly forgotton about that page

*Welcome to the nForum (nlabmeta)*. I re-doscivered it only after my recent related comment here.

- Discussion Type
- discussion topiccomorphism
- Category Latest Changes
- Started by varkor
- Comments 4
- Last comment by Urs
- Last Active Apr 16th 2023

- Discussion Type
- discussion topiclocally ringed topos
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Apr 16th 2023

added to locally ringed topos the characterization as algebras over the geometric theory of local rings.

I give pointers to two references that I know which say this more or less explicitly: Johnstone and Lurie. But I lost the page where Johnstone says this. I had it a minute ago, but then somebody distracted me, and now it is as if the paragraph has disappeared…

- Discussion Type
- discussion topicringed topos
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Urs
- Last Active Apr 16th 2023

brushed up ringed topos a little, added the version over any Lawvere theory and linked it to “related concepts” (for use at Tannaka duality for geometric stacks).

- Discussion Type
- discussion topiclocally ringed topological space
- Category Latest Changes
- Started by DavidRoberts
- Comments 30
- Last comment by Urs
- Last Active Apr 16th 2023

I was talking to an ex-Adelaide student now at Oxford about some technicalities they were trying to track down regarding locally ringed spaces. I checked locally ringed topological space, and found the Stacks Project reference was out of date. I replaced it with a link to the specific tag for the definition, at least.

- Discussion Type
- discussion topicringed space
- Category Latest Changes
- Started by DavidRoberts
- Comments 2
- Last comment by Urs
- Last Active Apr 16th 2023

- Discussion Type
- discussion topiccomplement
- Category Latest Changes
- Started by Todd_Trimble
- Comments 2
- Last comment by nLab edit announcer
- Last Active Apr 15th 2023

I fixed a mistake at complement (where it said complements are unique), and added more to Related Concepts.

- Discussion Type
- discussion topicElena Caviglia
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 15th 2023

- Discussion Type
- discussion topicdifferentiable stack
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Apr 15th 2023

I have added pdf-links to the reference

- David Carchedi,
*Categorical Properties of Topological and Diffentiable Stacks*, PhD thesis, Universiteit Utrecht, 2011 (dspace:1874/208971, pdf)

and promoted this to the top of the list, since I suppose this is the most comprehensive account that a reader might want to go to first. Will also edit accordingly at

*topological stack*- David Carchedi,

- Discussion Type
- discussion topicFrank Neumann
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 15th 2023

- Discussion Type
- discussion topicprincipal bundle
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Apr 15th 2023

I have added to

*principal bundle*a remark on their

*definition*As quotients;statements about (classes of) (counter-)examples of quotients

Thanks for pointers to the literature from this MO thread!

- Discussion Type
- discussion topicsphere fiber bundle
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Apr 15th 2023

added the statement that

The stable tangent bundle of a unit sphere bundle $S(\mathcal{V})$ in a real vector bundle $\mathcal{V} \overset{p}{\longrightarrow} M$ (Example \ref{UnitSphereBundles}) over a smooth manifold $M$ is isomorphic to the pullback of the direct sum of the stable tangent bundle of the base manifold with that vector bundle:

$T^{stab} S(\mathcal{V}) \; \simeq \; S(p)^\ast \big( T^{stab} M \oplus_M \mathcal{V} \big) \,.$Still need to add a more canonical reference and/or a proof.

- Discussion Type
- discussion topichigher observational type theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 10
- Last comment by nLab edit announcer
- Last Active Apr 15th 2023

- Discussion Type
- discussion topicAlonso Perez-Lona
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 15th 2023

- Discussion Type
- discussion topicorbifold
- Category Latest Changes
- Started by Urs
- Comments 48
- Last comment by Urs
- Last Active Apr 15th 2023

I am moving the following old query box exchange from orbifold to here.

old query box discussion:

I am confused by this page. It starts out by boldly declaring that “An orbifold is a differentiable stack which may be presented by a proper étale Lie groupoid” but then it goes on to talk about the “traditional” definition. The traditional definition definitely

**does not**view orbifolds as stacks. Neither does Moerdijk’s paper referenced below — there orbifolds form a 1-category.Personally I am not completely convinced that orbifolds are differentiable stacks. Would it not be better to start out by saying that there is no consensus on what orbifolds “really are” and lay out three points of view: traditional, Moerdijk’s “orbifolds as groupoids” (called “modern” by Adem and Ruan in their book) and orbifolds as stacks?

Urs Schreiber: please, go ahead. It would be appreciated.

end of old query box discussion

- Discussion Type
- discussion topicadditive functor
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Urs
- Last Active Apr 15th 2023

at

*additive functor*there was a typo in the diagram that shows the preservation of biproducts. I have fixed it.Also formatted a bit more.

- Discussion Type
- discussion topicÁngel Murcia
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 15th 2023

- Discussion Type
- discussion topicCarlos S. Shahbazi
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 15th 2023

- Discussion Type
- discussion topicordinary differential cohomology
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Apr 15th 2023

following public demand, I created an entry ordinary differential cohomology.

- Discussion Type
- discussion topicfree cocompletion
- Category Latest Changes
- Started by Urs
- Comments 33
- Last comment by varkor
- Last Active Apr 14th 2023

I was looking again at this entry, while preparing my category theory notes elsewhere, and I find that this entry is really bad.

With the co-Yoneda lemma in hand (every presheaf is a colimit of representables, and that is dealt with well on its page), the statement of free cocompletion fits as an easy clear Idea into 2 lines, and as a full proof in maybe 10.

The entry should just say that!

Currently the section “technical details” starts out right, but somehow forgets along the way what it means to write a proof in mathematics.

On the other hand, the section “Gentle introduction” seems to be beating about the bush forever. Does this really help newbies?

- Discussion Type
- discussion topicconservative cocompletion
- Category Latest Changes
- Started by varkor
- Comments 1
- Last comment by varkor
- Last Active Apr 14th 2023

- Discussion Type
- discussion topicAdS3-CFT2 and CS-WZW correspondence
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by Urs
- Last Active Apr 14th 2023

I have split off from

*holographic principle*and then expanded a good bit a few paragraphs on*AdS3-CFT2 and CS-WZW correspondence*together with a few commented references, trying to amplify how this case is given by an actual well-known theorem and at the same time seems to be generic for the more general cases of AdS-CFT which are currently much more vague.

- Discussion Type
- discussion topicmotivic Galois group
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by John Baez
- Last Active Apr 14th 2023

stated a kind of Idea/definiton at

*motivic Galois group*.Experts and experts-to-be, please check!

- Discussion Type
- discussion topicsuspended category
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by nLab edit announcer
- Last Active Apr 13th 2023

- Discussion Type
- discussion topicleft-determined model category
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by Thomas Holder
- Last Active Apr 12th 2023

- Discussion Type
- discussion topicBen Goertzel
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Apr 12th 2023

- Discussion Type
- discussion topicrewriting
- Category Latest Changes
- Started by David_Corfield
- Comments 10
- Last comment by zskoda
- Last Active Apr 12th 2023

I thought to add

- Amar Hadzihasanovic,
*Diagrammatic sets and rewriting in weak higher categories*, (arXiv:2007.14505)

- Amar Hadzihasanovic,

- Discussion Type
- discussion topiccreated limit
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by varkor
- Last Active Apr 11th 2023

- Discussion Type
- discussion topicsupply in a monoidal category
- Category Latest Changes
- Started by Mike Shulman
- Comments 3
- Last comment by Urs
- Last Active Apr 11th 2023

- Discussion Type
- discussion topicThe Stacks Project
- Category Latest Changes
- Started by Urs
- Comments 30
- Last comment by DavidRoberts
- Last Active Apr 11th 2023

created a “category: reference”-page The Stacks Project

I have only now had a closer look at this and am impressed by the scope this has. Currently a total of 2288 pages. It starts with all the basics, category theory, commutative algebra and works its way through all the details to arrive at algebraic stacks.

So besides my usual complaint (Why behave as if there are not sites besides the usual suspects on $CRing^{op}$ and either give a general account or call this

*The Algebraic Stacks Project*? ) I am enjoying seeing this. We should have lots of occasion to link to this. Too bad that this did not start out as a wiki.

- Discussion Type
- discussion topicdissipation
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Apr 10th 2023

- Discussion Type
- discussion topicHopf algebra
- Category Latest Changes
- Started by Urs
- Comments 24
- Last comment by zskoda
- Last Active Apr 10th 2023

added to the Properties-section at Hopf algebra a brief remark on their interpretation as 3-vector spaces.

- Discussion Type
- discussion topicterminal geometric morphism
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by jesuslop
- Last Active Apr 10th 2023

I keep wanting to point to properties of the terminal geometric morphism. While we had this scattered around in various entries (such as at

*global sections*, at*(infinity,1)-topos*and elsewhere – but not for instance at*(infinity,1)-geometric morphism*) I am finally giving it its own entry, for ease of hyperlinking.So far this contains the (elementary) proofs that the geometric morphism to the base $Set$/$Grpd_\infty$ is indeed essentially unique, and that the right adjoint is equivalently given by homs out of the terminal object.

- Discussion Type
- discussion topicWolfgang Rump
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 5
- Last comment by zskoda
- Last Active Apr 10th 2023

Created:

Wolfgang Rump is a mathematician working at the University of Stuttgart.

He got his PhD degree in 1978 from the University of Stuttgart, advised by Klaus Wilhelm Roggenkamp.

## Selected writings

- Wolfgang Rump,
*Almost abelian categories*, Cahiers de topologie et géométrie différentielle catégoriques, tome 42, no 3 (2001), p. 163–225. numdam.

- Wolfgang Rump,

- Discussion Type
- discussion topicSchwinger-Keldysh formalism
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Apr 10th 2023

- Discussion Type
- discussion topicvielbein
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by nLab edit announcer
- Last Active Apr 10th 2023

I have added to

*vielbein*a discussion of the vielbein as an orthogonal structure and, more in details as an example of a tiwsted differential c-structure for $\mathbf{c} : \mathbf{B} O(n) \to \mathbf{B} GL(n)$.

- Discussion Type
- discussion topicUrs Schreiber
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active Apr 10th 2023

- Discussion Type
- discussion topicIan Stark
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 9th 2023

- Discussion Type
- discussion topicbunched logic
- Category Latest Changes
- Started by spitters
- Comments 13
- Last comment by Urs
- Last Active Apr 9th 2023