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    • 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.

    • 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.)

    • 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.

    • 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 HilbHilb to one saying that they are the isomorphisms.

    • 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.

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

    • 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.

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

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

      Similarly, the 4-\psis 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 (3,4,6,10) and (4,5,7,11) are the first two steps in a longer sequence of sequences? I might expect another sequence (7,8,10,14) and (11, 12, 14, 18) corresponding to the fivebrane and the ninebrane. In other words, what happens when you look at n \times n-matrices with values in a division algebra for values of n 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 (\infty,1)-categorical exact, namely are fibration sequences/fiber sequences in an (\infty,1)-category of L_\infty-algebras.

      The cocycle itself is a morphism of L_\infty-algebras

      \mu : \mathfrak{siso}(n+1,1) \to b^2 \mathbb{R}

      and the extension it classifies is the homotopy fiber of this

      \mathfrak{superstring}(n+1,1) \to \mathfrak{siso}(n+1,1) \to b^2 \mathbb{R} \,.

      Forming in turn the homotopy fiber of that extension yields the loop space object of b^2 \mathbb{R} and thereby the fibration sequence

      b \mathbb{R} \to \mathfrak{superstring}(n+1,1) \to \mathfrak{siso}(n+1,1) \to b^2 \mathbb{R} \,.

      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 \infty-exact sequence here is to declare that the (\infty,1)-category of L_\infty-algebras is that presented by the standard modelstructure on dg-algebras on dgAlg^{op}. In there we can show that b \mathbb{R} \to \mathfrak{superstring} \to \mathfrak{siso} 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 \mathfrak{siso} and do the analog for \mathfrak{superstring} without really changing the nature of the problem, given how simple b \mathbb{R} 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

    • 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

    • 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!

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

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

      David Roberts

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

    • 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.

    • 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.

    • 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).

    • 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

    • 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...

    • 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.

    • 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.

    • 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).

    • 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.

    • 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.

    • 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.

    • 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.

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

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

    • 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?

    • 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

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