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- Discussion Type
- discussion topictrialgebra and Hopf monoidal category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 9th 2013

created entries

*trialgebra*and*Hopf monoidal category*also expanded the Tannaka-duality overview table (being included in related entries):

to contain the first entries of the corresponding “higher Tannaka duality” relations

- Discussion Type
- discussion topichopfish algebra
- Category Latest Changes
- Started by Urs
- Comments 23
- Last comment by Urs
- Last Active Apr 9th 2013

- Discussion Type
- discussion topicHilbert bimodule
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Apr 9th 2013

added to

*Hilbert bimodule*a pointer to the Buss-Zhu-Meyer article on their tensor products and induced 2-category structure.

- Discussion Type
- discussion topicrelative categories
- Category Latest Changes
- Started by Zhen Lin
- Comments 1
- Last comment by Zhen Lin
- Last Active Apr 8th 2013

I have now created relative category.

Question: Does the transferred model structure on $\mathbf{RelCat}$ resolve Rezk’s [2001] conjecture that the classification diagram of a model category is weakly equivalent to its simplicial localisation? The $N_\xi$ functor looks very close to computing the hammock localisation to me…

- Discussion Type
- discussion topicweak Hopf algebra
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 8th 2013

cross-linked

*weak Hopf algebra*and*fusion category*a bit more explicitly, added to both a reference to Ostrik’s article that shows the duality and added a corresponding item to

- Discussion Type
- discussion topichigher groupoid C*-algebras
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 8th 2013

I have added to

*groupoid convolution algebra*the beginning of an Examples-section titled*Higher groupoid convolution algebras and n-vector spaces/n-modules*.Conservatively, you can regard this indeed as just some examples of applications of the groupoid convolution algebra construction. But the way it is presented is supposed to be suggestive of a “higher C*-algebra” version of convolution algebras of higher Lie groupoids.

I have labelled it as “under construction” to reflect the fact that this latter aspect is a bit experimental for the moment.

The basic idea is that to the extent that we do have groupoid convolution as a (2,1)-functor

$C \colon Grpd \to Alg_{b}^{op} \simeq 2Mod$(as do do for discrete geometry and conjecturally do for smooth geometry), then this immediately means that it sends double groupoids to convolution sesquialgebras, hence to 3-modules with basis (3-vector spaces).

As the simplest but instructive example of this I have spelled out how the ordinary dual(commutative and non-co-commutative) Hopf algebra of a finite group arises this way as the “horizontally constant” double groupoid incarnation of $\mathbf{B}G$, while the convolution algebra of $G$ is the algebra of the “vertically discrete” double groupoid incarnation of $\mathbf{B}G$.

But next, if we simply replace the bare $Alg_b^{op} \simeq 2 Mod$ with the 2-category $C^\ast Alg_b$ of $C^\ast$-algebras and Hilbert bimodules between them and assume (as seems to be the case) that $C^\ast$-algebraic groupoid convolution is a 2-functor

$LieGrpd_{\simeq} \to C^\ast Alg_n^{op}$then the same argument goes through as before and yields convolution “$C^\ast$-2-algebras” that look like Hopf-C*-algebras. Etc. Seems to go in the right direction…

- Discussion Type
- discussion topiccohomological integration
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 7th 2013

seeing the announcement of that diffiety summer school made me think that we should have a dedicated entry titled

*cohomological integration*which points to the aspects of this discussed already elswhere on the nLab, and which eventually lists dedicated references, if any. So I created a stub.Does anyone know if there is a published reference to go with the relevant diffiety-school page ?

- Discussion Type
- discussion topicbibundle
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Apr 5th 2013

dropped some lines into a new Properties-section in the old and neglected entry

*bibundle*. But not for public consumption yet.

- Discussion Type
- discussion topicclosure operator
- Category Latest Changes
- Started by Urs
- Comments 24
- Last comment by Urs
- Last Active Apr 4th 2013

I felt we were lacking an entry

*closure operator*. I have started one, but don’t have more time now. It’s left in a somewhat sad incomplete state for the moment.

- Discussion Type
- discussion topicclassification of finite groups
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by zskoda
- Last Active Apr 4th 2013

just noticed that this morning some apparently knowledgable person signing as “Snoyle” added two paragraphs to finite group with technical details.

I have helped a bit with the syntax now and split off entries for quasisimple group and generalized Fitting subgroup

- Discussion Type
- discussion topicHopf C-star-algebra
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Apr 2nd 2013

started

*Hopf C-star algebra*(but my computer is running out of battery power now..)

- Discussion Type
- discussion topiccanonical Hilbert-space of half-densities
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 1st 2013

brief note:

*canonical Hilbert space of half-densities*

- Discussion Type
- discussion topicstar-algebras and dagger-categories
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 1st 2013

Added brief comments at

*star algebra*, at*dagger category*and at*category algebra*on how convolution algebras on dagger-categories are naturally star algebras.

- Discussion Type
- discussion topicTannakian category
- Category Latest Changes
- Started by adeelkh
- Comments 5
- Last comment by Tim_Porter
- Last Active Apr 1st 2013

I made a stub Tannakian category with some references.

- Discussion Type
- discussion topicLanglands dual groups and T-duality
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 31st 2013

Added the recent reference on Langlands dual groups as T-dual groups to both

*geometric Langlands correspondence*and*T-duality*together with a brief sentence. But nothing more as of yet.

- Discussion Type
- discussion topictopological algebra
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Mar 31st 2013

I could have sworn that we already had entries like “topological ring”, “topological algebra” or the like. But maybe we don’t, or maybe I am looking for the wrong variant titles.

I ended up creating a stub for topological algebra now…

- Discussion Type
- discussion topicC-star algebra
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Mar 31st 2013

I have added to C-star algebra the statement that the image of a $C^\ast$-algebra under an $\ast$-homomorphism is again $C^\ast$.

Also reorganized the Properties-section a bit and added more references.

- Discussion Type
- discussion topicEpsilons, epsilons, everywhere epsilons
- Category Latest Changes
- Started by TobyBartels
- Comments 3
- Last comment by TobyBartels
- Last Active Mar 30th 2013

- Discussion Type
- discussion topicgroupoid
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by David_Corfield
- Last Active Mar 29th 2013

the entry

*groupoid*could do with some beautifying.I have added the following introductory reference:

- Alan Weinstein,
*Groupoids: Unifying Internal and External Symmetry – A Tour through some Examples*, Notices of the AMS volume 43, Number 7 (pdf)

- Alan Weinstein,

- Discussion Type
- discussion topicmodules over Lie groupoid convolution algebras
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 29th 2013

I have started adding some references to

on modules ($C^\ast$-modules) of (continuous, etc..) convolution algebras of topological/Lie groupoids.

I still need to look into this more closely. A motivating question for this kind of thing is:

what’s the right fine-tuning of the definition of modules over twisted Lie groupoid convolution algebras such that for centrally extended Lie groupoids it becomes equivalent to the corresponding gerbe modules?

This seems fairly straightforward, but there are is some technical fine-tuning to deal with. I was hoping this is already stated cleanly in the literature somewhere. But maybe it is not. Or maybe I just haven’t seen it yet.

- Discussion Type
- discussion topiccentrally extended groupoid
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 29th 2013

Wrote a quick note at

*centrally extended groupoid*and interlinked a little, for the moment just motivated by having the link point somwhere.

- Discussion Type
- discussion topicGoldblatt-Thomason theorem
- Category Latest Changes
- Started by zskoda
- Comments 4
- Last comment by Tim_Porter
- Last Active Mar 29th 2013

- Discussion Type
- discussion topicfoliation of a Lie groupoid
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 25th 2013

- Discussion Type
- discussion topicLie algebroid-groupoid
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 25th 2013

- Discussion Type
- discussion topicfoliation of a Lie algebroid
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 25th 2013

am starting

*foliation of a Lie algebroid*

- Discussion Type
- discussion topicdouble Lie algebroid
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 25th 2013

stub for

*double Lie algebroid*

- Discussion Type
- discussion topicpolarization
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by crogers
- Last Active Mar 22nd 2013

as mentioned in another thread, I have expanded the Idea-section at

*polarization*in order to highlight the relation to*canonical momenta*(which I also edited accordingly).

- Discussion Type
- discussion topicPositive numbers
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Mar 22nd 2013

I keep making links to positive number, so now I filled them.

- Discussion Type
- discussion topicphase and phase space in physics
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Mar 22nd 2013

felt like making a terminological note on

*phase and phase space in physics*(and linked to it from the relevant entries).If anyone has more information on the historical origin of the term “phase space”, please let me know.

- Discussion Type
- discussion topicphase
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by TobyBartels
- Last Active Mar 22nd 2013

started a dismabiguation page for

*phase*. Feel invited to add further meanings.

- Discussion Type
- discussion topicCauchy's mistake
- Category Latest Changes
- Started by TobyBartels
- Comments 17
- Last comment by TobyBartels
- Last Active Mar 22nd 2013

- Discussion Type
- discussion topicsemiclassical state
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 21st 2013

Just in case you see me editing in the

*Recently Revised*list and are wondering:I have created and have started to fill some content into

*semiclassical state*. But I am not done yet and the entry is not in good shape yet. So don’t look at yet it unless in a mood for fiddling and editing.

- Discussion Type
- discussion topicclassical-to-quantum notions - table
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 21st 2013

I started an entry

*classical-to-quantum notions - table*for inclusion in “Related concepts”-sections in the relevant entries.This is meant to clean up the existing such “Related concepts”-lists. But I am not done yet with the cleaning-up…

- Discussion Type
- discussion topicsemiclassical+approximation
- Category Latest Changes
- Started by zskoda
- Comments 10
- Last comment by Urs
- Last Active Mar 21st 2013

New entry semiclassical approximation. It requires a careful choice of references. The ones at the wikipedia article are catastrophically particular, 1-dimensional, old and non-geometric and hide the story more than reveal. Stub Maslov index containing the main references for Maslov index.

- Discussion Type
- discussion topicGalois topos
- Category Latest Changes
- Started by Urs
- Comments 22
- Last comment by zskoda
- Last Active Mar 21st 2013

I created Galois topos following Dubuc’s article.

But I must be missing something about the notation: does it really mean to say that $A$ is an $\Delta Aut(A)$-torsor, as opposed to saying that it is

*associated*to an $\Delta Aut(A)$-torsor?

- Discussion Type
- discussion topicLagrangian dg-submanifold
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by Urs
- Last Active Mar 20th 2013

I have added the relations

coisotropic submanifold $\leftrightarrow$ Lagrangian submanifold in Poisson Lie algebroid

Dirac structure $\leftrightarrow$ Lagrangian submanifold in Courant Lie 2-algebroid

to

*Lagrangian submanifold*and cross-linked with various related entries, such as*polarization*.

- Discussion Type
- discussion topicEmpty entries
- Category Latest Changes
- Started by Tim_Porter
- Comments 16
- Last comment by TobyBartels
- Last Active Mar 20th 2013

There have been two empty pages created lately, both anonymous. They are at Riemann sphere and quasi inverse. It looks as if both were attempts to add something that was aborted.

- Discussion Type
- discussion topicwall crossing in Aarhus
- Category Latest Changes
- Started by Tim_Porter
- Comments 5
- Last comment by Tim_Porter
- Last Active Mar 20th 2013

This page, wall crossing in Aarhus, refers (in the future tense) to a course in 2010. The webpage link is broken as well. Does anyone have a link that could replace that one?

- Discussion Type
- discussion topicNew page: [[finitely generated object]]
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by IngoBlechschmidt
- Last Active Mar 19th 2013

A stub, so that I could link to it from the Café. Redirects from finitely presented object. (Is there any connection with finitely presentable object?) The general definition may not be the best, please check!

- Discussion Type
- discussion topicstrange entries
- Category Latest Changes
- Started by Tim_Porter
- Comments 2
- Last comment by TobyBartels
- Last Active Mar 19th 2013

Thought I would flag up that there have been two of these lately, ideal in semigroups and liars paradox. I waited to see if their ‘authors’ were going to come back and correct them, but so far they have not.

The entry Mochizuki's proof of abc is non-standard in form but has been updated by someone called Daniel.

- Discussion Type
- discussion topicPoisson tensor
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 18th 2013

created

*Poisson tensor*just for completeness, to be able to point to it from related entries.

- Discussion Type
- discussion topic(co)isotropic subspaces - table
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 18th 2013

created a simple table

*(co)isotropic subspaces - table*for inclusion in other entries, just so as to usefully cross-link the relevant entries

- Discussion Type
- discussion topicholonomy groupoid
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 15th 2013

Added a little to the Idea-section of

*holonomy groupoid*. But this deserves to be further expanded upon.

- Discussion Type
- discussion topicmodel structure on dg-algebras over an operad
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Mar 14th 2013

am starting model structure on dg-algebras over an operad

- Discussion Type
- discussion topicDiscrete event systems
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Mar 13th 2013

I created a stub for discrete event system as the grey link on tropical semirings was annoying me!

- Discussion Type
- discussion topic"The" category of sets
- Category Latest Changes
- Started by Todd_Trimble
- Comments 27
- Last comment by TobyBartels
- Last Active Mar 12th 2013

Mentions of the category $Set$ occur all over the nLab, but with quite a bit of plasticity of meaning. I thought it might be good to have another look at the entry Set and try to describe this plasticity as considered along various axes, to help readers who might be puzzled by “just what does the nLab think the category of sets is?” For example, one reads that the category of sets has marvelous properties such as being a well-pointed topos, and then a little further down one sees that $Set$ is not a topos according to predicative mathematics. This could be very confusing. Similarly, there are some pages in the nLab that assume $Set$ satisfies AC without batting an eye, while others discuss arcane weaker choice principles that $Set$

*might*satisfy. I think we need to be a just a bit more up-front about this, right on the page Set.In the definition section on Set, I made a meager start on this by declaring that the nLab adopts a ’pluralist’ position on the matter of sets and $Set$, and jotted down a few of the possible axes (“axises”, if I were James Dolan) of meaning and interpretation that guide how one thinks of $Set$, e.g., predicative vs. impredicative, classical vs. intuitionist, selection of choice principles, and others. I didn’t think really hard about this, but it might suggest useful ways of organizing the page.

I left out other axes such as “structural vs. material”, and said nothing about type theory. The page set talked more about this; I envision Set as concentrating more on properties of the category of sets.

I got to thinking about this when I began to wonder how Toby thinks about $Set$, which is maybe different from how I usually think about it. (Usually it feels slightly alien to me to posit say WISC as a possible choice principle for the

*category of sets*, which for me usually connotes a model of ETCS – normally I’d think of WISC instead as a possible axiom for a topos or a pretopos.) I was wondering whether Toby had a kind of “bottom line” for $Set$, say for example “$Set$ for me means at least a well-pointed topos with NNO, unless I choose to adopt a predicative mode”, or something like that. Anyway, discussion is invited.

- Discussion Type
- discussion topicAbsolute differential forms
- Category Latest Changes
- Started by TobyBartels
- Comments 32
- Last comment by TobyBartels
- Last Active Mar 10th 2013

After a few days’ editing, I’m announcing absolute differential form (using my neologism). This is a notion of differential form that can be integrated on a completely

*unoriented*submanifold. Examples from classical differential geometry include the arclength element on a Riemannian manifold and ${|\mathrm{d}z|}$ on the complex plane.Since there are classical examples, people must have thought about these before me, but I have never heard of them. Absolute differential forms are not linear (although they must satisfy a restricted linearity condition), and many typical examples are not smooth (although they are still continuous), so they don’t show up in the usual classification theorems. Has anybody heard of them before?

- Discussion Type
- discussion topicKostant-Souriau extension
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Mar 9th 2013

created brief remark at

*Kostant-Souriau extension*(beware the hyphen bug, sometimes only

*Kostant Souriau extension*will work, not sure why and when)(the hyphen bug combined with the cache bug combined with the low responsiveness make for a special experience…)

- Discussion Type
- discussion topicN-complex
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Mar 6th 2013

New stubs N-complex (the homological algebra where $d^N = 0$ with $N\gt 2$) and Michel Dubois-Violette. This interested me somewhat over a decade ago. Unfortunately, I missed the seminar talk yesterday in Zagreb by one of my colleagues, Pavle Pandžić, who found with his collaborator, very recently, that a more general and more insightful redefinition of Dirac cohomology, suggested by concrete applications in representation theory, involves the homological algebra of $N$-complexes. I hope there will be some writeup soon available.

- Discussion Type
- discussion topicLinear independence
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Mar 3rd 2013

I made linearly independent subset to satisfy a link. I put in something about the free-forgetful adjunction and something else about constructive mathematics.

- Discussion Type
- discussion topicsite
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Tim_Porter
- Last Active Feb 28th 2013

expanded a bit the discussion of morphisms of sites at site

- Discussion Type
- discussion topicn-groups
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by Mirco Richter
- Last Active Feb 26th 2013

There was a parity error at n-group; I fixed that and put in the low-dimensional examples.

- Discussion Type
- discussion topicArtin-Mazur codiagonal
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Tim_Porter
- Last Active Feb 24th 2013

I have created a diambiguation entry at Artin-Mazur codiagonal, as the old links at bisimplicial set lead to the entry on the codiagonal of a coproduct. I have used total simplicial set as the preferred term. Perhaps a more detailed discussion of this might be useful, but I have not got the time at the moment. (I am very slow at doing diagrams, :-( )

- Discussion Type
- discussion topicgeneric proofs and AC
- Category Latest Changes
- Started by Mike Shulman
- Comments 16
- Last comment by Mike Shulman
- Last Active Feb 22nd 2013

Wrote generic proof with some comments about a couple seemingly weaker versions of the axiom of choice that I've never seen mentioned anywhere before (has anyone else?). Toby and I noticed these a little bit ago while thinking about exact completions, but I just now realized that they're actually good for something: proving that the category of anafunctors between two small categories is essentially small (in the "projective" way).

- Discussion Type
- discussion topicTruth eventually and almost everywhere
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Feb 19th 2013

I wrote eventuality filter, although maybe this was unnecessary, and as it was mostly already there at net. Then I took some of the logic from there and adapted it to null set.

- Discussion Type
- discussion topicfree field theory
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Feb 18th 2013

created

*free field theory*with the formalization in terms of BV-complexes by Costello-Gwilliam.

- Discussion Type
- discussion topicThomason's homotopy colimit theorem
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Tim_Porter
- Last Active Feb 16th 2013

Does anyone know if we have a discussion, somewhere, of the theorem of Thomason linking homotopy colimits with Grothendieck constructions. I have looked in places that I thought were likely but found no trace of it, but sometimes things get buried in entries on other topics so are difficult to find.

- Discussion Type
- discussion topiccritics of string theory
- Category Latest Changes
- Started by zskoda
- Comments 60
- Last comment by Urs
- Last Active Feb 15th 2013

New entry critics of string theory to collect the references on controversies. I think they are often rambling and vague, not technically useful s the main references we want to collect under string theory and books in string theory. I have changed the sentence in string theory about mathematical definition of parts to somewhat more precise

But every now and then some aspect of string theory, some mathematical gadget or consequence found there is isolated and redefined independently and mathematically rigorously, retaining many features originally predicted.

The point is that most often one does not make rigorous the way some thing is defined via string theory, but one isolates an invariant of manifolds for example and defines a similar one via completely different foundations. The typical example is quantum cohomology which is defined in geometric terms and not in terms of field theory any more.

I have one disagreement with the entry: it says that topological quantum field theory has been discovered as part of string theory research, This is not true, TQFTs were found in 1977, 1978, 1980 articles of Albert Schwartz which had nothing to do with string theory. Only much later Atiyah’s formulation is influenced by string theory.

- Discussion Type
- discussion topicThe convenient setting of global analysis
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by TobyBartels
- Last Active Feb 13th 2013

I finally gave

*The convenient setting of global analysis*a*category: reference*-entry. Started adding pointers to it from the References-section of some relevant entries. But there will be many more left.

- Discussion Type
- discussion topicNew discussion stub: [[weak enrichment]]
- Category Latest Changes
- Started by TobyBartels
- Comments 7
- Last comment by Urs
- Last Active Feb 12th 2013

I moved some discussion from bicategory to weak enrichment, a new page. (Possibly it was already moved somewhere else, since Mike had already deleted it, but I couldn't find it.)