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- Discussion Type
- discussion topicsemisimple Lie algebra
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by TobyBartels
- Last Active Sep 7th 2010

started semisimple Lie algebra

- Discussion Type
- discussion topicTrivial stuff
- Category Latest Changes
- Started by TobyBartels
- Comments 4
- Last comment by TobyBartels
- Last Active Sep 6th 2010

trivial group, trivial ring, trivial Lie algebra, abelian Lie algebra (which may not be trivial exactly but by special dispensation is still too trival to be simple)

- Discussion Type
- discussion topichelp: typeset this proof
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Sep 6th 2010

there is a bunch of things on my to-do list concerning write-up of stuff on the nLab in the big context of my “diff cohomology in an (oo,1)-topos”-writeup. I am lagging behind. I could use some more help!

Of course you all are busy with your own stuff. But maybe there is a tiny chance somebody reads this here, maybe somebody who has been lurking all along, somebody who would enjoy helping out. I could say: I offer 60 reputation points! More seriously, this might be a chance to dip your feet into the water and join in to some interesting research. Here is the deal:

I have a LaTeX writeup of a fairly long proof that establishes the weak equivalence of 3 different strict 2-group models of the string 2-group. It appears as the proof of prop. 5.25 on p. 94 of these notes here. The trouble (for me) is that this proof involves some diagrams that would require code-cogs for implementation on the nLab. I want these diagrams on the nLab!

I have the LaTeX source code, of course, so it’s not much work to make this run on the nLab! But a bit of work. A tad more work than I find leisure for right now.

Let me know if you are interested and I’ll send you the LaTeX source!

Best, Urs

- Discussion Type
- discussion topicChern-Simons 2-gerbe
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Sep 6th 2010

I considerably expanded and reorganized the discussion at Chern-Simons 2-gerbe.

There is now a decvent Idea-section;

I created three subsections for three different kinds of constructions of this beast.

The first subsection contains now a detailed account of the consztruction by Brylinski-McLaughline of an explicit Cech-Deligne cocycle. I end this with saying that once the construction is there, proving that it is correct is easy. The mystery is maybe how one comes up with the construction in the first place!

So then in the next subsection I rederive this construction as a special case of the general methods described at infinity-Chern-Weil theory. So I show that from Lie integration of the underlying Lie algebra cocycle one gets a canonical lift to pseudo-connections with values in the Lie algebra, and turning the crank, out drops the Brylinski-McLaughlin construction. I’ll later see if I can streamline this discussion a bit more.

Then there is a third subsection which is supposed to deal with the construction of bundle 2-gerbe representatives. But here I am being lazy and just give the references so far. Even though the construction is actually simple.

- Discussion Type
- discussion topichomotopy category of an (infinity,1)-category
- Category Latest Changes
- Started by Harry Gindi
- Comments 2
- Last comment by Urs
- Last Active Sep 6th 2010

@Urs: I answered your query over at homotopy category of an (infinity,1)-category. The Ho(SSet) enrichment is given by the lax monoidal functor SSet -> Ho(SSet).

- Discussion Type
- discussion topicfork, split equalizer, codecogs
- Category Latest Changes
- Started by zskoda
- Comments 15
- Last comment by Eric
- Last Active Sep 6th 2010

Using codecogs recipe and ascii table I wrote short entries fork and split equalizer. For those who distinguish fork and cofork, I have hard time remembering which one is which one.

By the way, nForum is today having lots of problems on my computer, it asks for human recognition, it bails out my automatically remembered password many times out and resets the settings for markdown when writing etc. often.

- Discussion Type
- discussion topicMonoid object --> monoid
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by TobyBartels
- Last Active Sep 6th 2010

There was more about monoid objects at monoid than at monoid object, so I incorporated the latter into the former. (This means that the history of the latter is now at monoid object > history.

- Discussion Type
- discussion topicTemplate rearrangement
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Sep 5th 2010

I rearranged the template page so that the template came up top, on the grounds that this is what people will mostly want to copy and paste. Then they can scroll down for a more detailed example.

- Discussion Type
- discussion topicContributor update
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Sep 5th 2010

I've updated Contributors for this month. If there are any mistakes, I won't find them until October.

- Discussion Type
- discussion topicSimple objects: subobjects vs quotient objects
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by Todd_Trimble
- Last Active Sep 4th 2010

The definition at simple object referred to subobjects instead of quotient objects. Although these definitions are equivalent in abelian categories, it seems to me that we must use quotient objects to get the correct definition of a simple group, so I have changed it.

- Discussion Type
- discussion topicA composable pair is the stub of a commutative triangle.
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Sep 3rd 2010

Stubs at composable pair and commutative triangle

- Discussion Type
- discussion topicBaues-Wirsching cohomology
- Category Latest Changes
- Started by Tim_Porter
- Comments 1
- Last comment by Tim_Porter
- Last Active Sep 3rd 2010

I have added a bit of history to the entry on Baues-Wirsching cohomology. Whilst looking for something else I found a paper by Charlie Wells from 1979, extending the earlier ideas of Leech cohomology for semigroups to small categories. He defines various types of extension and classifies them using the same methods as B and W used a few years later.

- Discussion Type
- discussion topicChern-Simons form
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Sep 3rd 2010

am starting curvature characteristic form and Chern-Simons form.

But still working…

- Discussion Type
- discussion topicmatrix Lie group / matrix Lie algebra
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Sep 2nd 2010

started one-line stubs for matrix Lie group and matrix Lie algebra just so that the links to these, which started appearing at parallel transport and elsewhere do point somewhere

- Discussion Type
- discussion topic[skeleton]
- Category Latest Changes
- Started by DavidRoberts
- Comments 6
- Last comment by TobyBartels
- Last Active Sep 2nd 2010

Added a bit to skeleton about skeletons of internal categories

- Discussion Type
- discussion topicexact functor
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 30th 2010

added to exact functor a new subsection “Between abelian categories” and listed there (briefly) the standard characterizations of left/right exact functors in terms of preservation of left/right exact sequences.

Also added a reference by Michael Barr on the relation between exactness and respect for homology in very general contexts.

- Discussion Type
- discussion topicinjective object
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by TobyBartels
- Last Active Aug 27th 2010

added to injective object propositions and examples for injective modules and injective abelian groups

P.S. I am checking if I am missing something: Toën on page 48/49 here behaves as if it were clear that there is a model structure on positive cochain complexes of R-modules for all R in which the fibrations are the epis. But from the statements that i am aware of at model structure on chain complexes, in general the fibrations may be taken to be those epis that have injective kernels. For $R$ a field this is an empty condition and we are in business and find the familiar model structures. But for $R$ not a field? Notably simply $R = \mathbb{Z}$ What am I missing?

- Discussion Type
- discussion topicNew logic articles
- Category Latest Changes
- Started by TobyBartels
- Comments 6
- Last comment by Mike Shulman
- Last Active Aug 27th 2010

- Discussion Type
- discussion topic[over quasi-category]
- Category Latest Changes
- Started by Harry Gindi
- Comments 5
- Last comment by Harry Gindi
- Last Active Aug 26th 2010

I fixed the definition at over quasi-category so it makes the adjointness relationship clearer between overcategories and joins. In particular, Lurie’s notation and definition makes it very hard to see this. It’s much easier to see what’s going on when we look at things as follows: The join with $K$ fixed in the first coordinate, $S\mapsto i_K^{K\star S}: K\to K\star S$, where $i_K^{K\star S}$ is the canonical inclusion, is a functor $SSet\to (K\downarrow SSet)$. Then the undercategory construction gives the adjoint to this functor sending $(K\downarrow SSet) \to SSet$. This makes it substantially clearer to understand what’s going on, since $Hom_{SSet}(S, X_{F/}):= Hom_{K\downarrow SSet}(i_K^{K\star S}, F)$ is the set of those maps $f:K\star S\to X$ such that $f|K=f\circ i_K^{K\star S}=F$.

Lurie’s notation $Hom_{SSet}(S, X_{F/}):= Hom_F(K\star S, X)$ is nonstandard and inferior, since it obscures the obvious adjointness property.

The definition for overcategories is “dual” (by looking at the join of $K$ on the right).

- Discussion Type
- discussion topicnice simplicial topological space
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 25th 2010

started nice simplicial topological space with material provided by David Roberts

- Discussion Type
- discussion topicsimplicial topological space
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 24th 2010

stub for simplicial topological space

- Discussion Type
- discussion topicoo-Lie groupoid
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Aug 23rd 2010

considerably expanded Lie infinity-groupoid. But still stubby.

- Discussion Type
- discussion topicSomething about nothing
- Category Latest Changes
- Started by TobyBartels
- Comments 21
- Last comment by TobyBartels
- Last Active Aug 21st 2010

- Discussion Type
- discussion topicSifted colimits
- Category Latest Changes
- Started by TobyBartels
- Comments 3
- Last comment by TobyBartels
- Last Active Aug 20th 2010

Aleks Kissinger has given us sifted colimit. Although I don’t quite understand the definition.

- Discussion Type
- discussion topicSequential nets
- Category Latest Changes
- Started by TobyBartels
- Comments 3
- Last comment by TobyBartels
- Last Active Aug 20th 2010

I revisited some old discussion with Mike at sequence. Are you happy now, Mike?

- Discussion Type
- discussion topicJointly epimorphic families
- Category Latest Changes
- Started by TobyBartels
- Comments 5
- Last comment by TobyBartels
- Last Active Aug 19th 2010

I linked to it, so here is jointly epimorphic family, our newest stub.

- Discussion Type
- discussion topicoo-vector bundles
- Category Latest Changes
- Started by Urs
- Comments 36
- Last comment by zskoda
- Last Active Aug 18th 2010

started oo-vector bundle on my personal web, following my latest remarks in the thread here on deformation theory.

- Discussion Type
- discussion topicindecomposable object
- Category Latest Changes
- Started by FinnLawler
- Comments 17
- Last comment by Marc
- Last Active Aug 18th 2010

New page: indecomposable object, following (what I think is) Johnstone's definition. I also found it in some online topos theory lecture notes by Ieke Moerdijk and Jaap van Oosten.

Lambek and Scott give a different definition in

*Intro. to Higher-order Cat. Log.*, p. 168. I'm not sure how it relates to Johnstone's.I've also given a proof that indecomposable <=> connected in an extensive category. I'd be interested to know whether this hypothesis is the weakest possible, if anyone has any ideas (or just likely-looking references).

- Discussion Type
- discussion topicThin categories
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Aug 18th 2010

I could have sworn that we had something for thin category, at least a redirect, but we don’t. Or didn’t. Now we do.

Not much to it, just a note of terminology, like inductive limit or (0,1)-topos.

There’s also a diagram that I can’t to get to work there, if anybody wants to help.

- Discussion Type
- discussion topiccircle n-bundle with connection
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 18th 2010

created circle n-bundle with connection.

See the nForum thread on oo-Chern Weil theory for background.

- Discussion Type
- discussion topicGleason's Theorem for POVMs
- Category Latest Changes
- Started by TobyBartels
- Comments 4
- Last comment by Ian_Durham
- Last Active Aug 18th 2010

This is mathematically much simpler than the classical Gleason’s Theorem, but I added it to Gleason’s theorem anyway.

- Discussion Type
- discussion topicgluing from localizations
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by zskoda
- Last Active Aug 17th 2010

In gluing categories from localizations (zoranskoda) the main section

**From a family of localizations to a comonad**is fully rewritten in improved notation. In other way, it is explained better how to get a comonad from a cover of a category by not necessarily compatible flat localizations. This generalizes the Sweedler's coring to relative situations. Now from such data one can make a two category, which I will explain in few days.

This is a preliminary to something I am writing at the moment namely to explain in such terms actions of comonads and monoidal categories on such descent categories. This part will be analogous to description of equivariant maps among G-manifolds in pairs of local charts, but because of the distributive laws with coherences, the thing complicates.

- Discussion Type
- discussion topicstacked cover
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 17th 2010

started stacked cover

- Discussion Type
- discussion topicQuantification
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by TobyBartels
- Last Active Aug 17th 2010

I started quantifier, but I ran out of time to say all that I wanted. I’ll probably get back to it in a couple of hours.

- Discussion Type
- discussion topicContents
- Category Latest Changes
- Started by TobyBartels
- Comments 2
- Last comment by Mike Shulman
- Last Active Aug 17th 2010

I redid everything that includes contents using the new click-based menu system. This includes HomePage; there didn’t seem to be a need anymore to have two columns, so I put them back in one column. However, those are separate issues; we could put them back in two columns again and still do the click thingy.

- Discussion Type
- discussion topicStates in statistical physics
- Category Latest Changes
- Started by TobyBartels
- Comments 3
- Last comment by Ian_Durham
- Last Active Aug 16th 2010

I added some stuff about states in statistical physics to state.

- Discussion Type
- discussion topicStates of operator algebras
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Aug 16th 2010

I moved states in AQFT and operator algebra to state on an operator algebra, and I also included there a redirect from state on an algebra, which had been requested on quantum state. Hopefully it makes sense that way.

I almost wrote state on an algebra myself, but I’m glad that I didn’t, since everything that I would have said (and more) was already on the extant page.

- Discussion Type
- discussion topicpossible spam: [PrimeDeGold] and [theory of primes]
- Category Latest Changes
- Started by DavidRoberts
- Comments 23
- Last comment by TobyBartels
- Last Active Aug 16th 2010

A couple of new pages have appeared, theory of primes and PrimeDeGold, the latter being the author of the former (and itself). They seem like nonsense, but perhaps someone (Andrew? Toby?) is doing testing? Or else testing for a sort of spamming?

- Discussion Type
- discussion topicnonabelian Lie algebra cohomology
- Category Latest Changes
- Started by zskoda
- Comments 6
- Last comment by Urs
- Last Active Aug 16th 2010

- I added word "abelian" when the article Lie algebra cohomology talks about Lie algebra cohomology as cohomology of CE complex. While one can combine this abelian cohomology with various actions to get nonabelian information, there is also a genuine nonabelian cohomology of Lie algebra as it is for Lie groups. At least some special cases are understood in low dimensions. For example the problem of extensions of Lie algebras by nonabelian Lie algebras leads to 1,2,3 nonabelian cocycles; 2-cocycles are analogues of factor systems. I will create some entry about that, and put a longer exposition in "private" web following my old notes from Spring/Summer 1997. For start I just started a stub nonabelian Lie algebra cohomology.

- Discussion Type
- discussion topicoo-Lie algebra cohomology
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Aug 16th 2010

created and expanded infinity-Lie algebra cohomology.

There is now a section on the $(\infty,1)$-topos theoretic interpretation, and one on how to understand $\infty$-Lie algebra extensions as special cases of the general nonsense on principal $\infty$-bundles.

The discussion leaves quite a bit of room for polishing, but I don’t feel like spending much more time on this right this moment.

- Discussion Type
- discussion topic[shrinkable map]
- Category Latest Changes
- Started by DavidRoberts
- Comments 3
- Last comment by DavidRoberts
- Last Active Aug 16th 2010

shrinkable map, so I can reference it for Urs’ question on Hurewicz fibrations.

- Discussion Type
- discussion topicOne is the loneliest article.
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active Aug 16th 2010

- Discussion Type
- discussion topicAdded new links to my nLab page
- Category Latest Changes
- Started by Stephen Britton
- Comments 3
- Last comment by Stephen Britton
- Last Active Aug 16th 2010

I added new links to my nLab page. I have links to my ’table of categories’ and ’database of categories’ here

- Discussion Type
- discussion topicBaues-Wirsching cohomology
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by Todd_Trimble
- Last Active Aug 15th 2010

Baues-Wirsching cohomology of small categories got a stub. Updates to Hans-Joachim Baues.

New entry Masoud Khalkhali related to the quite recent cyclic cohomology entry (which still lacks the basic material).

- Discussion Type
- discussion topicadded an example to "examples" in monad
- Category Latest Changes
- Started by jcmckeown
- Comments 9
- Last comment by TobyBartels
- Last Active Aug 15th 2010

- a description of a free monad on Sets in Cat, the successor monad, and remarks.

- Discussion Type
- discussion topicFeynman proof
- Category Latest Changes
- Started by zskoda
- Comments 4
- Last comment by Ian_Durham
- Last Active Aug 14th 2010

In my personal $n$lab a bibliography for an interesting topic, which some of my peers in Zagreb got recently interested in: Feynman proof of the Lorentz force equations (zoranskoda). It is funny – deriving gauge theories and even gravity just from commutation relations for the generators (coordinates and “covariant momenta”) without any action principle, that is without assuming Lagrange or Hamilton formalism to hold. It has nice extensions and it may be important for the philosophy of gauge theories aka connections on vector bundles. Any ideas of categorification may be interesting…

When talking personal $n$lab I wrote a long general advice page for students who may ask for my future mentorship in Zagreb of some sort. Well, if I stay in Zagreb. The things are getting rough for science here, and I am not sure of my own future.

- Discussion Type
- discussion topicSati Geometric...M-branes
- Category Latest Changes
- Started by zskoda
- Comments 5
- Last comment by zskoda
- Last Active Aug 11th 2010

Hisham Sati in January posted a survey Geometric and topological structures related to M-branes. In a hard effort of several hours of intense work I created an entry containing hyperlinked bibliography of that article (I took LaTeX source, scraped off various LaTeX commands like bf, it, bibitem etc. and then started creating various hyperlinks). Most of the hyperlinks to the arxiv and few to the project euclid are created so far. Many items still do not have proper external links which would be very welcome. This is a very nice bibliography for something of much interest to Urs, me and some other nlabizants, and I would like to have it practical for our systematic online study.

- Discussion Type
- discussion topicPasting lemmas
- Category Latest Changes
- Started by FinnLawler
- Comments 3
- Last comment by FinnLawler
- Last Active Aug 10th 2010

Added a proof of the pasting lemma to pullback, and the corresponding lemma to comma object (also added the construction by pullbacks and cotensors there).

- Discussion Type
- discussion topicInfinitary Lawvere theories
- Category Latest Changes
- Started by TobyBartels
- Comments 15
- Last comment by TobyBartels
- Last Active Aug 7th 2010

I’ve started infinitary Lawvere theory following this n-Forum discussion.

- Discussion Type
- discussion topicNew entry on Bourn plus query on semi-Abelian categories
- Category Latest Changes
- Started by Tim_Porter
- Comments 5
- Last comment by zskoda
- Last Active Aug 4th 2010

I have added a stub entry to the lab on Dominique Bourn. There are quite a few links that need developing there as the protomodular category stuff is quite rudimentary. I would need to learn more about it to fill things up so if anyone does feel they can help, please charge ahead.

- Discussion Type
- discussion topic3-Lie
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by zskoda
- Last Active Aug 3rd 2010

am starting to create stubs

and am heading for

- Discussion Type
- discussion topicQuestion about typed varietes of algebras
- Category Latest Changes
- Started by Andrew Stacey
- Comments 4
- Last comment by TobyBartels
- Last Active Aug 2nd 2010

On variety of algebras appears the sentence “(This paragraph may be original research. Probably the concept does appear in the literature but under a different name.)”. The paragraph in question is about

*typed*varieties of algebras. Looking at the history, this sentence (and indeed, the whole page!) appears to be due to Toby (Bartels).I’m curious as to what part that sentence refers to, in particular due to my interest in what I call

*graded*varieties of algebras (nomenclature coming from algebraic topology and*graded*cohomology theories), which I thought was just an example of a*heterogeneous*variety of algebras, a term that I’ve come across in the literature. Certainly the concepts feel closely related, and it took a fair amount of paper chasing to find the term “heterogeneous” (though “many-sorted” theories seemed a bit more of a common term), but despite my interest, I’m no expert and am sure I’m missing something. Problem is: I don’t know what and I don’t know how to properly formulate my question!

(Added in edit): Actually, I see that the term “multisorted” is in use on Lawvere theory.

- Discussion Type
- discussion topic(hyperconnected,localic)
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Aug 2nd 2010

- Discussion Type
- discussion topicsubnet
- Category Latest Changes
- Started by Todd_Trimble
- Comments 4
- Last comment by TobyBartels
- Last Active Aug 2nd 2010

The definition of “subnet” under net looked wrong to me (part of the wrongness was obvious), so I changed it so that it looks correct to me. Could someone please give independent verification?

- Discussion Type
- discussion topicProper and improper subsets
- Category Latest Changes
- Started by TobyBartels
- Comments 4
- Last comment by TobyBartels
- Last Active Jul 30th 2010

I found a good constructive definition of proper subset and put it in there. Also I wrote improper subset.

Edit: also family of subsets; see below.

- Discussion Type
- discussion topicdifferential cohomology - contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jul 29th 2010

started floating TOC differential cohomology - contents

- Discussion Type
- discussion topicdiploma higher gauge theory
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by zskoda
- Last Active Jul 29th 2010

A semester ago I announced a possible mentorship in Zagreb if a physics student would like to take to digest and write a diploma on the basis of Baez-Schreiber work on higher gauge theory. Nobody chose the topic but the page in my personal nlab is left out from those times, and maybe it will be recycled by a future announcement, though it is questionable as I am likely to leave my present institution in few months. But in order to be functional, it is good to have also the list of literature which i just compiled, including the appended list of very advanced references so that it might serve at all levels. Suggestions and usage for your own purposes are welcome.

- Discussion Type
- discussion topicholonomy
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by zskoda
- Last Active Jul 29th 2010

added a

*stub*entry for holonomy.Just the bare definition, and of that even only the most naive one. Don’t have time for more. But created it anyway because I needed the link.

(Sounds a bit like like: I was young and needed the money…)

- Discussion Type
- discussion topicThe *real* real numbers have characteristic zero!
- Category Latest Changes
- Started by TobyBartels
- Comments 19
- Last comment by zskoda
- Last Active Jul 28th 2010

I wrote about generaliasations of real numbers and managed to follow one link to create characteristic.

- Discussion Type
- discussion topicautomorphism oo-Lie algebra
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jul 28th 2010

started automorphism infinity-Lie algebra with some basics from Sullivan’s old article