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    • 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

    • I have to admit that I simply cannot parse many of the entries on type theory and related.

      Now, this is certainly my fault, as I am not spending any considerable time to follow this. But on the other hand my impression is that many statements here are not overly complicated, and that I ought to be following at least roughly what's going on. But I don't.

      One thing is that when I try to look up precise definitions such as at type theory I run into long pieces of text. I am not sure what to make of this.

      My understanding was at some point that all of type theory is really just another way of speaking about categories. Instead of "object" A we say "type" A. Instead of morphism p : U \to A we say p : A " p is of type A" and the like.

      Can we have some Rosetta-stone entry where all the type-theoretic language is translated into plain category theory this way?

      For instance I am suspecting that what is going on at identity type is somehow another way of saying equalizer. But I am not sure. Can anyone help me?

    • This comment is invalid XML; displaying source. <p>motivated by Domenico's <a href="http://www.math.ntnu.no/~stacey/Vanilla/nForum/comments.php?DiscussionID=905&page=1#Item_40">latest comment</a> I copied the material on Whitehead towers in (oo,1)-toposes from the end of <a href="https://ncatlab.org/nlab/show/universal+covering+space">universal covering space</a> into a dedicated entry:</p> <ul> <li><a href="https://ncatlab.org/nlab/show/Whitehead+tower+in+an+%28infinity%2C1%29-topos">Whitehead tower in an (infinity,1)-topos</a></li> </ul>
    • Due to popular demand (well, maybe not) I have uploaded my presentation to the APS March Meeting from Friday. It can be found here. I linked it from the bottom of the quantum channel page.
    • Based on a discussion I had with someone after my talk today, I tossed an idea up on the entanglement page concerning how to use categories to model the process of entangling something which I think could be extremely useful to physicists. But it needs a bit of work and I have a plane to catch. I will note that the idea came to me during the conversation when I recalled p. 36 in Steve Awodey's book.

    • polished and expanded the Idea-section at AQFT

    • Zoran,

      concerning your paper with Durov and the sheaf category defined on p. 22, I am wondering:

      it would almost seem as if something essentially equivalent is obtained if we would very slightly change the definition of the site (Rings with a chosen nilpotent ideal) and think of it as the tangent category of the category of rings, i.e. of Mod, thought of as being the category of square-0-extensions of rings.

      So I am suggesting that we look at sheaves on (the opposite of) Mod

      Do you see what I mean?

    • Why the pluralized title in cochains on simplicial sets, unlike in the rest of nlab ? In addition the second plural "on simplicial sets" is misleading, as if it we were talking about cochains defined on a collection of simplicial sets, rather than cochains on a single simplicial set.

      Typoi discussoin, collectoin...

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