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May 27, 2010

nLab Digest

Posted by David Corfield

I wanted to give people an idea of what has been going on in recent weeks at the very active nLab and associated personal wikis, so I asked members to describe some of the important contributions of late. Before I relay these, remember that anyone is welcome to participate in this great collaborative project, and also to join in discussions at nForum.

Let’s hear first from Urs Schreiber.

Urs Schreiber: Apart from the seminar notes on (,1)(\infty,1)-topos theory that I keep writing (but I think (,1)(\infty,1)-category theory - contents and (,1)(\infty,1)-topos theory - contents are beginning to look good) I am mainly trying to bring on my personal web the entry differential cohomology in an (,1)(\infty,1)-topos, and the entries relating to it, into shape.

Progress is unfortunately much slower than I would hope. But some things improve, slowly but surely. For instance I am glad that I finally cleaned up and started to expand Lie infinity-algebroid with some serious indications of and examples for the incarnation as infinitesimal \infty-Lie groupoids. In that context I also started telling more details of the story at Lie infinity-groupoid, but that is still pretty stubby.

In the course of this I keep adding bits and pieces here and there. For instance this morning I posted some proofs at model structure on simplicial presheaves on the Localization of (,1)(\infty,1)-presheaves at a coverage (instead of at a full Grothendieck topology). This I am using in the discussion at locally n-connected (n,1)-topos as a tool for constructing examples of those. The central motivating example is the category of (,1)(\infty,1)-sheaves on the site CartSp equipped with its good open cover coverage – and finally I had recently found the time to leave some comments at CartSp as to the relevance of this innocent-looking little category.

All this, of course, is still an outgrowth of the development we had a while ago at homotopy groups in an \infty-topos. After some back and forth it looks like the following generalization of the picture there is beginning to stabilize finally: \infty-Lie theory is about geometric homotopy theory in a locally \infty-connected \infty-topos, but not with respect to the global geometric morphisms HΠGrd\mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\leftarrow}{\to}} \infty Grd, but with respect to a relative such morphisms HΠ infH redΠGrd\mathbf{H} \stackrel{\overset{\Pi_{inf}}{\to}}{\stackrel{\leftarrow}{\to}} \mathbf{H}_{red} \stackrel{\overset{\Pi}{\to}}{\stackrel{\leftarrow}{\to}} \infty Grd, where H\mathbf{H} is an infinitesimal thickening of H red\mathbf{H}_{red}, which in turn is a finite thickening of the point.

The picture emerging here is still being drawn at structures in an \infty-topos.

I just wish the day had more hours. It’s shame how long I have been elaborating on all this already, and still counting.

Andrew Stacey: My recent n-Activity can be roughly split into three areas. The first of these is differential topology of mapping spaces, the second is related to topological vector spaces, whilst the third is not so much nLab as nForum.

Sparked by a project with David Roberts to mollify his construction of the topological bigroupoid, I started transferring my old notes on the differential topology of loop spaces to the nLab. This is something I’ve been threatening to do for a while, and finally the overwhelming urge of sheer laziness prompted me to do something about it. (To explain, I found that I was writing out essentially the same argument for the bigroupoid as I had for the loop space; as any good mathematician knows, there are only three numbers: 0, 1, and ∞. Having started writing the details twice, I was worried that I would end up writing them out ∞ number of times. So writing them into the nLab means that I can accomplish that ∞ number of times in one go.) In order to make it look something like active research, I’ve extended it from loop spaces to mapping spaces. The case where the mapping space is a compact (smooth) manifold is known, so I’ve had to go to sequentially compact Froelicher spaces to find something new to say.

(NB For those unfamiliar with terms from functional analysis, a mollifier is a function that takes non-smooth maps and makes them smooth.)

This led me into the demesne of topological vector spaces, as the linear models of mapping spaces can be fairly arbitrary locally convex topological vector spaces. Again, there’s a fair amount of material that I’ve been meaning to import to the nLab but not gotten round to. I thought I’d start with Greg Kuperberg’s amazing graph. With Tim van Beek, I’ve been having fun reminding myself of the subtleties and differences between barreled spaces, bornological spaces and lots of other variants.

Finally, I’ve been working on improving the nForum. There, we now have a browser-savvy mathematics capability that will serve you your mathematics to your taste: the chef’s special is, of course, MathML but if you prefer it raw we can do Math Tartar (aka, the raw itex source), or if you prefer it flame-grilled to avoid any risk of Mad-Mathematician Disease, we can also do PNG. For those with exotic tastes, we can also offer it as SVG. We also have a host of new features: tags, a separate area for discussions that aren’t particularly nLab focussed, a notice board, and one day soon we may even get our own favicon.

Tim van Beek: Urs has created a template page.

There was some activity concerning general relativity and Lorentzian geometry, starting with smooth Lorentzian space.

A remark of Zoran Skoda let me to study hyperfunctions a little bit, which is about a very neat alternative approach to generalized functions, more general than Schwartz distributions, and at least in one dimension simpler, too. The link to D-modules is still missing, however.

My quest to get some classical AQFT content to the nLab has led to Haag-Kastler axioms, Haag-Kastler vacuum representation, Wightman axioms, the Osterwalder-Schrader theorem and a definition in terms of these of conformal field theories. This definition includes operator product expansions, which provide the link to vertex operator algebras. As a side effect some operator theory got added, too, for example on the pages von Neumann algebra, operator algebra and unbounded operator. The ultimate goal is of course to join the dots with category theory, for example it would be nice to have the proof of Müger of the Doplicher-Roberts reconstruction theorem on the nLab.

Todd Trimble: Most of my recent efforts have been small scale, but two of the larger ones were AT category and composition algebra. I’ve also been adding at times to the document on Surface Diagrams on my personal web.

Mike Shulman: Recently I’ve been working mostly on two pages. One is codiscrete cofibration, which is about a way to canonically recover a proarrow equipment from a 2-category with sufficient structure. Essentially, one can define in 2-categorical terms which cospans ought to be the cographs of profunctors, and then construct a proarrow equipment out of those. The literature doesn’t seem to contain a version of this which works for arbitrary enriched categories (only for Cauchy-complete ones), but I think that by a judicious choice of a factorization system (now described at codiscrete cofibration it can be made to work. A neat side benefit of this is that there is actually an intrinsic characterization of the VV-fully-faithful functors in 2-categorical terms in VCatV Cat (they are not just the representably fully faithful ones).

I have also been thinking about how functorial this construction is, which also doesn’t seem to be in the literature. It seems right now to be very functorial; in fact it may even be a full embedding of some 3-category of 2-categories into some 3-category of proarrow equipments. This is particularly interesting because a number of monads on equipments relative to which one might want to define generalized multicategories can be obtained in this way from 2-monads on 2-categories, which require less work to construct.

The other circle of pages I’ve been working on is derivators, along with pointed derivators and stable derivators. I haven’t been able to find a really satisfying introductory treatment of derivators anywhere, so I’m gradually trying to create one at the nLab as I puzzle things out myself. Working with derivators is an interesting experience; it feels kind of like being a new category theory student again and having to learn a whole new set of tools. Statements that are “obvious” for 1-categories require entirely new proofs in the world of derivators, and I’m only gradually getting comfortable with the tools used in those proofs. Hopefully writing slow and detailed proofs on the nLab pages will help me and others learn to manipulate those tools as easily as we do the more familiar toolbox of 1-categories.

David Roberts: I’ve been thinking a little bit about partitions of unity. Covers with subordinate partitions of unity (numerable covers) are particularly nice, but are only guaranteed to exist for paracompact spaces, or normal spaces if you are handed a locally finite cover. In fact numerable covers form a Grothendieck pretopology, and it is using this site that all the classical theorems for algebraic topology should be stated, instead of just for paracompact spaces with the usual open cover pretoplogy.

Actually it is enough, instead of having a partition of unity, to have an arbitrary collection of functions u α:X 0u_\alpha:X \to \mathbb{R}_{\geq 0} such that αu α(x)\sum_\alpha u_\alpha(x) is finite for all xx, because this can be further refined to end up with a subordinate partition of unity. This is essentially an old result from a PhD thesis from the 60s. This can be interpreted as requiring XX having enough coprobes with values in the rig 0\mathbb{R}_{\geq 0}

Zoran Skoda has also been busy.

Interested now in contributing? Find out more about nLab, and how easy it is to contribute.

Posted at May 27, 2010 10:18 AM UTC

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20 Comments & 1 Trackback

Re: nLab Digest

Nice to see people doing all these things! Maybe someone can say what Joyal has been up to lately on his “CatLab”.

I’m using my personal page on the nnLab to write a paper on electrical circuits based on some ideas in This Week’s Finds. But you shouldn’t bother looking at it yet because it’s very preliminary. I’ll announce it when it’s closer to ready. I just wanted to let you know that I’m not completely slacking off when it comes to nnLab work.

Posted by: John Baez on May 27, 2010 3:53 PM | Permalink | Reply to this

Joyal’s CatLab

Maybe someone can say what Joyal has been up to lately on his “CatLab”.

Impressive chapters keep appearing and growing. In reverse order of appearance:

Quillen model structures

The natural model structure on CatCat

Distributors and barrels

Factorization systems

Weak factorization systems.

There are also “sociological” entries, such as

North American School of Category Theory

Posted by: Urs on May 27, 2010 5:34 PM | Permalink | Reply to this

Re: nLab Digest

There are also other developments that didn’t get mentioned in the above. One day when I have some other things out of the way I want to talk about the nnForum discussions I had with Domenico Fiorenza on DW-theory and related.

Domenico’s student Giuseppe Malavolta is currently beginning to write his master thesis on the nnLab, working title being:

The standard model of particle physics: a dictionary for the mathematician.

Work takes place on this page in Domenico’s personal area, while discussion about it goes on here.

Posted by: Urs Schreiber on May 27, 2010 6:17 PM | Permalink | Reply to this

Re: nLab Digest

The link above (Zoran Škoda is “busy”) to the entry in nForum where I post less important reports of my updates to nlab is not very representative; when I find some topic more imrpotant I create a separate nForum therad for it. The most important topics I tried to make alive this Spring have been a circle of topics related to the modern geometric and categorical picture of differential operators like crystals, D-modules and so on, cf.

I am especially fond of the deep contribution of Japanese school (of Michio Sato and his follower Kashiwara) microlocal analysis which combines analysis of pseudodifferential operators with sheaf theoretic picture of D-modules, perverse sheaves and a version of generalized functions called hyperfunctions. Microlocal pertains to keeping the local information also of the tangent direction, thus combining the Fourier approach to the generalizations of differential operators with ideas in symplectic geometry and geometric ideas steming from geometrical optics and semiclassical approximation in quantum mechanics.

Most of the activity related to mathematical physics in nlab is about the best kind of QFTs matehmatically - topological ones, with supersymmetry. The semiclassical approximation in such cases is exact, that is subject to a phenomenon of localization of the Feynman integral which is in turn related to the equivariant localization in topology. Being educated as a physicist I am not too excited about topological theories; one wants indeed some dynamics. Now we all know that beyond topological game, one stucks in difficult analytic issues like renormalization. It looks like completely different kind of formalisms than those favorite in nlab about TQFTs. So my reasoning is that when the semiclassical approximation is not exact, the first term in the expansion is typically still of topological nature, hence could be interpreted in terms of the same ideas like in TQFT case. Thus the semiclassical methods (and all the machinery above for its establishment like microlocal analysis, cohomological tools like Maslov index and so on) could possibly give a hint how to extend the categorical methods (in a geometrically sensitive way) from TQFTs to more general situation.

So the overall direction is to combine geometry and category theory in study of quantization of general field theories.

Posted by: Zoran Skoda on May 28, 2010 11:41 PM | Permalink | Reply to this

Re: nLab Digest

Although the nForum began life as merely a place to record latest changes to the nLab, with the recent technological improvements Andrew has made, I believe the nForum is now ready to become an important member of the nFamily.

We have the nCafe, which is like the nCommunity newspaper. A predefined set of journalists, with occasional guest editorials, write articles and generate interesting comments.

We have the nLab, which is like the nCommunity research library, complete with stacks and study rooms where active projects take place.

And now we have the nForum. This is really more like what the nCafe was envisioned to be. It is the place to come and casually discuss maths, physics, and philosophy mostly from the nPOV. The nForum now has the same itex capability as both the nCafe and the nLab. The “Source” feature allows you to copy and paste other’s itex commands and these can be pasted directed into the nLab. Hence, the nForum is now an effective way to generate content for the nLab.

As an illustration of some examples of the nForum put to very good use, see the very fascinating discussion between Urs Schreiber and Domenico Fiorenza:

and more recently

I also imagine a day where the nForum and the category theory mailing list develop a closer relationship, i.e. imagine the category theory mailing list with full itex capability. That is what the nForum is now capable of becoming.

Posted by: Eric Forgy on May 29, 2010 2:09 AM | Permalink | Reply to this

Re: nLab Digest

Eric wrote:

This is really more like what the nnCafe was envisioned to be.

I think the passive voice — “was envisioned” — conceals the fact that different people had different visions of the nnCafe. My vision was not much like the nnForum. But that’s no big deal: as long as a bunch of people are enjoying the nnForum, it’s a good thing.

Posted by: John Baez on May 29, 2010 3:20 PM | Permalink | Reply to this

Re: nLab Digest

True true. What I mean is the nForum is actually more cafe-like than the nCafe.

Posted by: Eric Forgy on May 29, 2010 5:16 PM | Permalink | Reply to this

Re: nLab Digest

It’s not the metaphor I use, but it has a sense to it. Even though the regular clientele is not too large, the place can sometimes get noisy, so that you either have to put on your headphones while you work at your laptop, or have to ask the others to please pipe down on occasion (as it’s not really a cafe, it’s a workplace!).

To me the Lab still feels sort of cool and underground, but not far from the Lab is the Forum which is a de facto tea room where you can hear constant chatter and the click-click of chalks on the blackboards. It’s a great workplace though, thanks to the collective spirit that is developing there.

Math Overflow is a lot more bustling. I feel there really needs to be greater synergy between the Lab and MO: in one direction, I see a lot of wisdom at MO which could and should be properly archived at the Lab (especially because I also see a great deal of duplicated effort at MO). It’s going to take some time.

Posted by: Todd Trimble on May 29, 2010 6:50 PM | Permalink | Reply to this

Re: nLab Digest

I see a lot of wisdom at MO which could and should be properly archived at the Lab (especially because I also see a great deal of duplicated effort at MO).

The MO folks could use Wikipedia for this, don’t they do that? Wikipedia already is a great source for math, the reason that I do not contribute to Wikipedia is that I think for every topic that I would like to write about there are thousands of people who could do it better than me…

BTW: The nLab seems to be rather prominent in google searches, the page about the Kochen-Specker theorem is no. 15 on the list for “Kochen-Specker”.

Posted by: Tim van Beek on May 30, 2010 8:22 AM | Permalink | Reply to this

Re: nLab Digest

Wikipedia is a godsend for the scientific researcher, no question about it, and the math entries there dwarf anything we have so far at nLab. Of course you’re right, Tim: people at MO use and cite WP all the time. (I also see them cite nLab a gratifyingly large number of times.)

I still believe that nLab has a unique value and actually does a better job than WP on some topics (and has stuff WP doesn’t have). Something to keep in mind is that Wikipedia is based on the encyclopedia model and therefore has self-imposed constraints, whereas we at the nLab have much more freedom. For example, you won’t find all that many proofs at WP. Distilling insights into clear proofs and having them openly available is a tremendous benefit of the nLab project. Let freedom ring!

(I’ll have to check out Kochen-Specker. Google often personalizes searches, so that different people get different results.)

Posted by: Todd Trimble on May 30, 2010 1:03 PM | Permalink | Reply to this

Re: nLab Digest

Google often personalizes searches…

What?! For this they need to collect data about my recent searches…which would be illegal if my employer did it (privacy protection is very important in Germany).

Wikipedia is based on the encyclopedia model and therefore has self-imposed constraints

Yes, that is another reason why I do not like to contribute to it.

…stuff WP doesn’t have…

Like, for example, an entry about the Bisognano-Wichmann theorem. This result concerns the relation between the modular operators associated with certain space-time regions of the Minkowski space-time in the vacuum representation. It’s older than I am, but, suprisingly, it seems that it took a long time until Tomita-Takesaki theory became a standard tool for people in AQFT. I’m sure that there is some interesting history.

BTW: Urs has asked for a spin-statistics page on the nLab (and a PCT page should go hand in hand with it), here are the best sources for the Haag-Kastler approach that I know, any further recommendations are welcome:

Borchers, Yngvason: “On the PCT-Theorem in the Theory of Local Observables”

Daniele Guido, Roberto Longo: “An Algebraic Spin and Statistics Theorem” (Dedicated to Hans Borchers on the occasion of his 70th birthday).

There is of course JB’s Spin, Statistics, CPT and All That Jazz, quote: “Maybe sometime I’ll explain this part and finish writing up a nice little spin-statistics FAQ”.

I guess that has not come to pass yet, has it?

Posted by: Tim van Beek on May 30, 2010 2:05 PM | Permalink | Reply to this

Re: nLab Digest

Google often personalizes searches…

What?! For this they need to collect data about my recent searches… which would be illegal if my employer did it (privacy protection is very important in Germany).

What can I say? I thought it was well-known that Google does this, certainly in the United States, although I can’t say how it operates in Germany. I would find it interesting to know more about this.

(I only heard just recently – was it from Toby? – that there’s now a rival search engine called Scroogle which arose particularly from worries over privacy issues.)

Posted by: Todd Trimble on May 30, 2010 3:07 PM | Permalink | Reply to this

Re: nLab Digest

There is a cultural clash between google, facebook and the understanding of privacy protection in Europe, the German ministry of consumer protection plans to propose a law later this year which is supposed to force google not to save any personalized data. How they intend to achieve that I do not know.

I’m not really troubled, in fact I would like to be observerd more closely, maybe we could get some of the people over at the CIA or NSA interested in category theory :-)

Well, in Germany that would be either the Verfassungsschutz if someone thinks I am a danger for the country, or the Bundesnachrichtendienst, which is the German equivalent of the CIA. Hey, they always look for people who are good in math and with computers, you can’t get fired, they pay well and they moved to Berlin recently (they were located in a small town near Munich before that). Maybe if we start talking about cracking RSA with category theory I will be offered a job :-)

Posted by: Tim van Beek on May 30, 2010 3:34 PM | Permalink | Reply to this

Re: nLab Digest

They don’t need much more than your IP to do some personalizing. Also, use google.com and google.de from the same IP - generally you’ll get different responses, particularly if you turn off cookies. For example, in my experience if one searches for math articles, the responses are usually biased towards articles by authors whose nationality matches a. the google domain, e.g. google.de or google.it, b. the nationality of your IP.

Posted by: Bobo on May 31, 2010 10:54 AM | Permalink | Reply to this

Re: nLab Digest

For those of us too ancient to keep up with all these multiple fora, such summary/reviews from time to time
on the primary blog (this one for me) are very helpful

Posted by: jim stasheff on May 30, 2010 4:03 PM | Permalink | Reply to this

Re: nLab Digest

While the above discussions on wikipedia etc. is worthy and interesting and nlab awareness in general is important, the main purpose of conceived This Month’s Finds in nlab eventually born as nlab-digest is (at least in spirit of our discussions few months ago) to attract attention and new contributors to a range of topics we work in nlab, and which may not be reflected on cafe. In particular, it came as a response to my complaints that e.g. physics is underrepresented. To make the nlab articles more staring into eyes we invested time into creating various tables of contents (cf. e.g. this one), literature lists and so on. Sometimes we go into automatizing access to things on the net we often need. For example, the page

is (for now) a partially hyperlinked list of references to Hisham Sati’s survey article concerning one of the central topics of our interest. So we do everything to make one conglomerate of interesting areas in our focus easier to work and collaborate on.

Posted by: Zoran Skoda on May 31, 2010 7:04 PM | Permalink | Reply to this

Re: nLab Digest

There is now a stub about the spin-statistics theorem and about the PCT theorem.

There are many heuristic explanations in the framework of Lagrangian QFT, and there are the proofs in the Wightman approach, what I am looking for now are simple and general proofs in the Haag-Kastler approach (the references that I do know are already mentioned in the reference sections in the nLab).

Is it true, as Borchers writes, that the PCT theorem was first proved in 2000 using the correspondence of the modular objects of the wedge algebras with the representation of the Poincare group as discovered by Bisognano and Wichmann in 1975?

(We need something like stackoverflow for physics.)

Posted by: Tim van Beek on May 31, 2010 9:43 PM | Permalink | Reply to this

Re: nLab Digest

We need something like stackoverflow for physics.

There is something very curious about online fundamental physics discussion: it tends not to work.

Is it a coincidence that most Wikipedia articles on math are between decent and remarkable, while most Wikipedia articles on fundamental physics are between half-baked and weird?

Is it a coincidence that there are quite a few weblogs with good discussion of math but hardly ever a good discussion of fundamental physics on any weblog? (There are some weblogs with very good physics entries , but the comment section discussion hardly every becomes constructive.)

I have some observation on why this is so, but don’t want to get into it here. For the moment I just want to say:

I think it is good that people with genuine deep understanding like you currently have in the nnLab a quiet area to produce decent entries on fundamental physics, without having to fight the usual smoke bombs thrown around that leave behing nothing but a great confusion. The physics contents of the nnLab is still very young (in terms of man-years spent on it), much younger still than the also still very young math content, but I quietly enjoy seeing it develop slowly but surely to something nice and good.

Let’s keep going this way for a while.

Posted by: Urs Schreiber on May 31, 2010 10:10 PM | Permalink | Reply to this

Re: nLab Digest

Tim wrote:

Is it true, as Borchers writes, that the PCT theorem was first proved in 2000 using the correspondence of the modular objects of the wedge algebras […] ?

I’m not sure I understand your question. Various versions of the PCT theorem have been around for many decades. So if your question was:

Is it true, as Borchers writes, that the PCT theorem was first proved in 2000 (using the correspondence of the modular objects of the wedge algebras […])?

then the answer is a resounding NO, and I urge you to look at this:

  • R. F. Streater and A. S. Wightman, PCT, Spin and Statistics, and All That, reprinted by Addison-Wesley, New York, 1989.

which proves this theorem using the Garding-Wightman axioms, based on earlier ideas by Fermi and others.

But if your question was:

Is it true, as Borchers writes, that the PCT theorem was first proved using the correspondence of the modular objects of the wedge algebras […] in 2000?

then the answer is a resounding MAYBE. I.e.: this may have been the first time that particular technique was used to prove this result.

By the way, there’s a little heuristic discussion of the PCT theorem at the end of this.

Posted by: John Baez on June 2, 2010 6:23 PM | Permalink | Reply to this

Re: nLab Digest

Various versions of the PCT theorem have been around for many decades.

Yes…agreed…

That’s why I wrote

There are many heuristic explanations in the framework of Lagrangian QFT, and there are the proofs in the Wightman approach…

Both Wightman’s book and your page are mentioned on the spin statistics theorem page.

…the answer is a resounding maybe…

The answer of Borchers and Yngvason is a resounding “first done here” in the theory of local observables, On the PCT-Theorem in the Theory of Local Observables and my question was and is:”Is that true?”, did no one prove PCT in the theorey of local observables before?

“theorey of local observables” refers to the set of axioms one uses, not to the method of proof.

Posted by: Tim van Beek on June 3, 2010 6:37 PM | Permalink | Reply to this
Read the post Jet Categories at the nForum
Weblog: The n-Category Café
Excerpt: nLab activity
Tracked: October 21, 2013 10:31 AM

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