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- Discussion Type
- discussion topicAnswer to the query over at structured (infinity,1)-topos?
- Category Latest Changes
- Started by Harry Gindi
- Comments 73
- Last comment by Mike Shulman
- Last Active Apr 27th 2010

Mike left a query box over at structured (infinity,1)-topos about admissibility structures. I am pretty sure that the admissibility structure is not, as the statement in the article says, a grothendieck topology. Rather, it is a class of morphisms that is in some way compatible with the grothendieck topology. At least looking at Toën's notes (which it seems are essentially a version of HAG II restricted to ordinary categories and ordinary stacks (I'm not positive that this is fully accurate, but I'm reasonably confident in the statement)), a geometric structure is a class of morphisms that is compatible with the grothendieck topology satisfying a number of conditions (that seem to match the axioms for an admissibility structure given here!). Correct me if I'm wrong, but it appears that an admissibility structure is precisely the class of morphisms P in the definition of a geometric context (or maybe even the pair (τ,P)).

Here's the link. Anyway, if this is true, it appears to answer Mike's question (once suitably generalized to (∞,1)-categories).

If I'm mistaken, please let me know.

I've put this in the (Latest Changes) category because at the moment, there is no nLab general category.

- Discussion Type
- discussion topiclocal addition
- Category Latest Changes
- Started by Andrew Stacey
- Comments 3
- Last comment by Andrew Stacey
- Last Active Apr 27th 2010

Created local addition to contain the definition and some useful auxiliary stuff. Took a little out of smooth loop space as a seed (for some reason, the extraction got mangled but I think I got it right in the end.)

- Discussion Type
- discussion topicHelp me! I'm trying to understand Bakalov and Kirillov
- Category Latest Changes
- Started by Bruce
- Comments 17
- Last comment by Bruce
- Last Active Apr 27th 2010

- I started a page titled <a href="http://www.ncatlab.org/nlab/show/Help+me!+I%27m+trying+to+understand+Bakalov+and+Kirillov">Help me! I'm trying to understand Bakalov and Kirillov</a> for those (like me) who need help in understanding some of the calculations in this book. I posted my first question. Will try to ask people to answer it.

- Discussion Type
- discussion topicmodel
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 27th 2010

I felt we needed a dedicated entry on model/category of models. So I started one. But just a puny stub so far.

- Discussion Type
- discussion topicsequential colimit
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Apr 26th 2010

ahm, another stupid question: what should sequential colimit point to? directed colimit?

- Discussion Type
- discussion topicunitary morphism
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Apr 26th 2010

I slightly expanded unitary morphism. In particular I added the example of unitary operators.

Then at unitary operator I in turn added the definition in terms of unitary morphisms. I also changed the former link to adjoint to a link to Hilbert space adjoint (since the former points to the categorical notion of adjoint). Also I changed the sentence saying that the unitary operators are the automorphisms in $Hilb$ to one saying that they are the isomorphisms.

- Discussion Type
- discussion topicnatural isomorphism
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 26th 2010

edited natural isomorphism a bit more

- Discussion Type
- discussion topicdagger-categories
- Category Latest Changes
- Started by John Baez
- Comments 2
- Last comment by Eric
- Last Active Apr 26th 2010

In preparation for week296 I corrected the definition of dagger-compact category, since it was missing some coherence laws. The most convenient way to include these was to add a page containing Selinger's definition of symmetric monoidal dagger-category . This in turn forced me to add pages containing definitions of associator, unitor, "braiding":http://ncatlab.org/nlab/show/braiding and unitary morphism. Some of my links between these pages are afflicted by the difficulty of getting daggers to appear in names of pages. Maybe a lab elf can improve them.

Hmm, html links didn't work here, so I'm trying textile.

- Discussion Type
- discussion topicorientation in generalized cohomology
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 24th 2010

- Discussion Type
- discussion topicZeidler
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by zskoda
- Last Active Apr 24th 2010

I have moved the personal data on Eberhard Zeidler from QFT entry to his own new-created entry.

- Discussion Type
- discussion topic[[Dmitry Tamarkin]]
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Apr 23rd 2010

Dmitry Tamarkin and Gonçalo Tabuada; 2 new references at microlocal analysis.

- Discussion Type
- discussion topicMayer-Vietoris sequence
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 22nd 2010

cerated at fibration sequence an Examples-subsection on the special case of Mayer-Vietoris sequences. From the nPOV, where it becomes a triviality, of course.

- Discussion Type
- discussion topichomotopy BV-algebra
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Apr 20th 2010

After having received an email from Bruno Valette I have now at least added a minimum of references to the stub entry homotopy BV-algebra.

- Discussion Type
- discussion topicGaudin model
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by zskoda
- Last Active Apr 20th 2010

Gaudin integrable model as a special case of Hitchin integrable system, and expansion of entry Branislav Jurčo.

- Discussion Type
- discussion topicpseudoalgebras
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Apr 19th 2010

Created pseudoalgebra for a 2-monad as a place to store something I just noticed about normalization, and a related stub at isocofibration.

- Discussion Type
- discussion topicunbounded operator(s)
- Category Latest Changes
- Started by Tim_van_Beek
- Comments 4
- Last comment by zskoda
- Last Active Apr 19th 2010

I made a first draft of a page about unbounded operators, the battle plan contains some basic definitions, explanation of some subtleties of domain issues and what it means to be affiliated to a von Neumann algebra. Right now, only the rigged Hilbert space page refers to it.

- Discussion Type
- discussion topicGray-categories
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Apr 19th 2010

I added some examples to Gray-category, including also a non-example which has fooled several people.

- Discussion Type
- discussion topicDivision algebras and supersymmetry II
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by zskoda
- Last Active Apr 16th 2010

The nCafé is currently haunted by a bug that prevents any comments from being posted. This should eventually go away, hopefully. For the time being I post my comment in reply to the entry Division Algebras and Supersymmetry II here:

Thanks, John and John for these results. This is very pleasing.

The 3-s rule implies that the Poincaré superalgebra has a nontrivial 3-cocycle when spacetime has dimension 3, 4, 6, or 10.

Similarly, the 4-s rule implies that the Poincaré superalgebra has a nontrivial 4-cocycle when spacetime has dimension 4, 5, 7, or 11.

Very nice! That's what one would have hoped for.

Can you maybe see aspects of what makes these cocycles special compared to other cocycles that the Poincaré super Lie algebra has? What other cocycles that involve the spinors are there? Maybe there are a bunch of generic cocycles and then some special ones that depend on the dimension?

Is there any indication from the math to which extent and are the first two steps in a longer sequence of sequences? I might expect another sequence and corresponding to the fivebrane and the ninebrane. In other words, what happens when you look at -matrices with values in a division algebra for values of larger than 2 and 4?

Here a general comment related to the

**short exact sequences**of higher Lie algebras that you mention:properly speaking what matters is that these sequences are -categorical exact, namely are fibration sequences/fiber sequences in an -category of -algebras.

The cocycle itself is a morphism of -algebras

and the extension it classifies is the homotopy fiber of this

Forming in turn the homotopy fiber of that extension yields the loop space object of and thereby the fibration sequence

The fact that using the evident representatives of the equivalence classes of these objects the first three terms here also form an exact sequence of chain complexes is conceptually really a coicidence of little intrinsic meaning.

One way to demonstrate that we really have an -exact sequence here is to declare that the -category of -algebras is that presented by the standard modelstructure on dg-algebras on . In there we can show that is homotopy exact by observing that this is

*almost*a fibrant diagram, in that the second morphism is a fibration, the first object is fibrant and the other two objects are almost fibrant: their Chevalley-Eilenberg algebras are almost Sullivan algebras in that they are quasi-free. The only failure of fibrancy is that they don't obey the filtration property. But one can pass to a weakly equivalent fibrant replacement for and do the analog for without really changing the nature of the problem, given how simple is. Then we see that the sequence is indeed also homotopy-exact.This kind of discussion may not be relevant for the purposes of your article, but it does become relevant when one starts doing for instance higher gauge theory with these objects.

Here some further trivial comments on the article:

Might it be a good idea to mention the name "Fierz" somewhere?

page 3, below the first displayed math: The superstring Lie 2-superalgebra is [an] extension of

p. 4: the bracket of spinors defines [a] Lie superalgebra structure

p. 6, almost last line: this [is] equivalent to the fact

p. 13 this spinor identity also play[s] an important role in

p. 14: recall this [is] the component of the vector

- Discussion Type
- discussion topiclocally presentable category
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by Urs
- Last Active Apr 15th 2010

I tried at locally presentable category to incorporate the upshot of the query box discussions into the text, then moved the query boxes to the bottom

- Discussion Type
- discussion topic(oo,1)-category of (oo,1)-presheaves
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Apr 15th 2010

added to (infinity,1)-category of (infinity,1)-presheaves a section on how it is the free (oo,1)-cocompletion

- Discussion Type
- discussion topic(oo,1)-category of (oo,1)-functors
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 15th 2010

added statement to (infinity,1)-category of (infinity,1)-functors on how limits and colimits are computed objectwise - here.

- Discussion Type
- discussion topiccoherence theorems
- Category Latest Changes
- Started by Todd_Trimble
- Comments 3
- Last comment by Todd_Trimble
- Last Active Apr 15th 2010

- As further outgrowth spurred by Sridhar's work on anafunctors and cliques: a stubby start on coherence theorems.

- Discussion Type
- discussion topicCategorified Symmetries
- Category Latest Changes
- Started by zskoda
- Comments 16
- Last comment by zskoda
- Last Active Apr 14th 2010

- Why I can not have this

1. Introduction 3

1.1 Categories and generalizations . . . . . . . . . . . . . . . . . 3

1.2. Basic idea of descent . . . . . . . . . . . . . . . . . . . . . . 5

2. From noncommutative spaces to categories 5

2.1. Idea of space and of noncommutative space . . . . . . . . . 5

2.2. Gel’fand-Naimark . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3. Nonaffine schemes and gluing of quasicoherent sheaves . . . 6

2.4. Noncommutative generalizations of QcohX . . . . . . . . . . 6

2.5. Abelian versus 1-categories . . . . . . . . . . . . . . . . . . 7

but instead I have automatic numbers like 1,2,3, 5, 8 (I know why, but how to avoid it??). I do NOT want nlab to make it like word, I want my numbering to stay it is, and if possible keeping the paragraphs. Putting > for quotation did not help!

- Discussion Type
- discussion topiclocalization of an (oo,1)-category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 14th 2010

edited localization of an (infinity,1)-category a bit. But more needs to be done.

- Discussion Type
- discussion topiclocally small (oo,1)-category
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Apr 14th 2010

- Discussion Type
- discussion topicclosed time-like curves
- Category Latest Changes
- Started by Ian_Durham
- Comments 46
- Last comment by zskoda
- Last Active Apr 12th 2010

- Added an entry on closed time-like curves. Also edited a few things in physicscontents in order to clean it up a bit.

- Discussion Type
- discussion topicmainstream mathematics
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by zskoda
- Last Active Apr 11th 2010

I have created in my own nlab, a programmatic entry mainstream mathematics (zoranskoda).

- Discussion Type
- discussion topicLie algebroid theory
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by zskoda
- Last Active Apr 10th 2010

in reply to discussion on the blog I

added more details to Lie algebroid

added a reference by Courant to Lie algebroid, Poisson Lie algebroid and tangent Lie algebroid

created Legendre transformation as a placeholder that currently just serves to keep some references on Legendre transformation from the point of view of Lie algebroid theory.

- Discussion Type
- discussion topicautonomous double categories
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Apr 10th 2010

I have started to work out a slew of definitions at autonomous double category. Work in progress.

- Discussion Type
- discussion topicobservable
- Category Latest Changes
- Started by zskoda
- Comments 8
- Last comment by Tim_van_Beek
- Last Active Apr 9th 2010

New entry observable wanted at many entries. For now the very basic stuff, with a view toward maximal generality.

- Discussion Type
- discussion topicsoliton
- Category Latest Changes
- Started by zskoda
- Comments 3
- Last comment by zskoda
- Last Active Apr 9th 2010

- Discussion Type
- discussion topicThesis finished!
- Category Latest Changes
- Started by Guest
- Comments 16
- Last comment by Tim_Porter
- Last Active Apr 9th 2010

- I have put a copy of my thesis, as it is being handed up for examination, on my personal web.

David Roberts

- Discussion Type
- discussion topicunbiased composites in double categories
- Category Latest Changes
- Started by Mike Shulman
- Comments 6
- Last comment by Eric
- Last Active Apr 8th 2010

Added some comments about the possibility of 2-dimensional unbiased composites in double categories.

- Discussion Type
- discussion topiccoinductive Trimble oo-categories
- Category Latest Changes
- Started by Mike Shulman
- Comments 26
- Last comment by DavidRoberts
- Last Active Apr 8th 2010

I added some comments to Trimble n-category regarding a coinductive way to state the definition, which I think is very clean and neat, and also a little mind-blowing.

- Discussion Type
- discussion topicIan: latest changes
- Category Latest Changes
- Started by Ian_Durham
- Comments 10
- Last comment by Andrew Stacey
- Last Active Apr 7th 2010

- As per someone's suggestion, I am going to do what Zoran does and have one, continuously running forum topic with my latest changes.

My latest change: added response to Zoran's query at quantum mechanics.

- Discussion Type
- discussion topiccontributors to algebraic geometry
- Category Latest Changes
- Started by zskoda
- Comments 4
- Last comment by zskoda
- Last Active Apr 7th 2010

While Prof. Joyal in his joyalscatlab keeps the lists of contributors to category theory and to homological algebra, I thought it might be useful to have some complementary list for 2-3 other fields in our nlab. But this could be too much work. So I restrained with contributors to algebraic geometry, as over 20 are alerady present in the nlab anyway (algebraic geometry has much being intertwined with the development of category theory since Serre and Grothendieck). Created Jean-Pierre Serre and Shigeru Mukai, wanted also at the Timeline entry; and David Mumford. This entry should be useful as we do not have top entry of contents for alegbraic geometry. (and I do not plan it soon).

- Discussion Type
- discussion topicDonaldson-Thomas invariant
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by zskoda
- Last Active Apr 7th 2010

Idea at Donaldson-Thomas invariant, hopefully correct to zeroth approximation, but not sure. Related update of Dominic Joyce.

- Discussion Type
- discussion topicquasi-category theory contents
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 7th 2010

created floating toc quasi-category theory contents

there is still a lot left to type in concerning quasi-category theory, but it seemed to me about time to collect what is there and organize it, so that one can see what we have, what is missing, and so that readers find their way to their information

- Discussion Type
- discussion topicspam at factorization algebra
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Eric
- Last Active Apr 7th 2010

some Anonymous Coward added an unmotivated link to some web-hosting site at factorization algebra. I have rolled back the entry to before that edit.

I am wondering if the creatures who do such things wiill at least eventually learn something from reading about things like factorization algebras...

- Discussion Type
- discussion topicgeneralized multicategories
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Apr 6th 2010

I have been greatly expanding the entry generalized multicategory, to work up gradually from the most explicit and easy-to-understand Leinster definition through profunctors, double categories, and up to the most general situation considered in my paper with Geoff, with examples. There's lots more left to do, but I have to run to a seminar now.

- Discussion Type
- discussion topicGrothendieck's existence theorem
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Apr 6th 2010

New entry Grothendieck existence along with many redirects. New person entry Steven Kleiman. Pdf link to write up of Gerd Faltings' 1995 Bonn lectures at moduli space of bundles and some new reference here and there.

- Discussion Type
- discussion topicDold fibrations
- Category Latest Changes
- Started by DavidRoberts
- Comments 1
- Last comment by DavidRoberts
- Last Active Apr 6th 2010

I fixed up an obvious error in a claim at Dold fibration: the counterexample there showed the converse of what was claimed (and I think the error was due to me). Now I'd like to find a proper counterexample, so I'll ask on MathOverflow, in case anyone has one up their sleeve.

- Discussion Type
- discussion topichom-object in a quasi-category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Apr 5th 2010

created hom-object in a quasi-category

- Discussion Type
- discussion topicGrothendieck fibration
- Category Latest Changes
- Started by Mike Shulman
- Comments 5
- Last comment by Tim_Porter
- Last Active Apr 5th 2010

Did some reorganizing at Grothendieck fibration.

- Discussion Type
- discussion topicSVG Editor HowTo
- Category Latest Changes
- Started by Andrew Stacey
- Comments 12
- Last comment by Andrew Stacey
- Last Active Apr 4th 2010

Created SVG Editor HowTo whilst trying to convert the codecogs monstrosity at exercise in groupoidification - the path integral to SVG (conversion happening in the Sandbox for the moment, and definitely not finished yet).

- Discussion Type
- discussion topicA partial addition and some added details
- Category Latest Changes
- Started by Ian_Durham
- Comments 3
- Last comment by Ian_Durham
- Last Active Apr 3rd 2010

- I began working on a page for the GNS construction but started dozing off at my keyboard (so it needs someone to help finish it). Also I added a note on QBism and category theory to the page on quantum states. In it I assert that category theory is, in essence, a theory about relations (bear with me on this one - read the entry).

- Discussion Type
- discussion topicquantum state
- Category Latest Changes
- Started by Ian_Durham
- Comments 12
- Last comment by Ian_Durham
- Last Active Apr 3rd 2010

- Added page quantum state.

- Discussion Type
- discussion topic‘folk’
- Category Latest Changes
- Started by TobyBartels
- Comments 4
- Last comment by Mike Shulman
- Last Active Apr 2nd 2010

I added a brief note to folk model structure to reflect the other terms suggested on another thread.

- Discussion Type
- discussion topicmonoidal bicategories
- Category Latest Changes
- Started by Mike Shulman
- Comments 4
- Last comment by Mike Shulman
- Last Active Apr 2nd 2010

Started monoidal bicategory with a bunch of references to the tortuous literature on braided and symmetric things. Please correct me if I left anything out. It would be nice to have the "accepted/correct" definitions here eventually, for reference.

- Discussion Type
- discussion topicSome updates
- Category Latest Changes
- Started by Ian_Durham
- Comments 7
- Last comment by Ian_Durham
- Last Active Apr 2nd 2010

- Didn't have much free time, but managed to add a little bit to Stokes phenomenon and responded to Zoran's query about Birkhoff's theorem. I suggested we rename it "Birkhoff-von Neumann theorem" so as not to confuse it with other similarly titled theorems.

- Discussion Type
- discussion topicGroup Presentation
- Category Latest Changes
- Started by Tim_Porter
- Comments 2
- Last comment by zskoda
- Last Active Apr 1st 2010

I have started an entry on group presentations. This will lead to a discussion of identities among relations as well as feeding into the rewriting page.

- Discussion Type
- discussion topicNew page: [[proper subset]]
- Category Latest Changes
- Started by TobyBartels
- Comments 41
- Last comment by Mike Shulman
- Last Active Mar 31st 2010

Not much there, but I created it: proper subset.

- Discussion Type
- discussion topicClassical field theory
- Category Latest Changes
- Started by Ian_Durham
- Comments 3
- Last comment by Ian_Durham
- Last Active Mar 29th 2010

- I added a query to classical field theory.

- Discussion Type
- discussion topic"Graphical" quantum channels
- Category Latest Changes
- Started by Ian_Durham
- Comments 9
- Last comment by Eric
- Last Active Mar 29th 2010

- I broke out a very rough page on what I'm tentatively calling graphical quantum channels. If anyone has some time or interest, it's really rough and I could use some help further developing this. Essentially, my aim is to develop a fully robust theory of quantum channels in category theoretic terms, but with an eye for pedagogy and simplicity.

- Discussion Type
- discussion topicmultisymplectic geometry
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by zskoda
- Last Active Mar 29th 2010

created multisymplectic geometry by effectively reproducing a useful survey website (see references given). But added a few wrapping sentences on the nLab perspective

- Discussion Type
- discussion topicMenagerie
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Tim_Porter
- Last Active Mar 27th 2010

I have given a list of chapters and section headings for the Menagerie notes (first 10 chapter).

- Discussion Type
- discussion topicpartially ordered dagger category
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active Mar 27th 2010

there have been recent edits at partially ordered dagger category. i edited a bit in an attempt to polish.

Tim Porter mentions parially ordered groupoids here. I am not sure why. These are not dagger categories, are they? This should go in another entry then, I suppose?

- Discussion Type
- discussion topiccartesian morphism
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active Mar 26th 2010

added to cartesian morphism

in the section for ordinary categories the definition in terms of pullbacks of over-categories

in the section on (oo,1)-categories more details on the definition and a very useful equivalent reformulation

- Discussion Type
- discussion topicregular singular point
- Category Latest Changes
- Started by zskoda
- Comments 1
- Last comment by zskoda
- Last Active Mar 25th 2010

The basic notion (though in fact nontrivial) regular singular point needed at Fuchsian equation. Redirects irregular singular point.

- Discussion Type
- discussion topicKapranov's NCG
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by zskoda
- Last Active Mar 25th 2010

New entries Kapranov's noncommutative geometry and Dmitry Kaledin. New references and typographic improvements at noncommutative algebraic geometry.