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    • I’ve started porting my notes “differential topology of loop spaces” over to the nlab, starting at differential topology of mapping spaces. As part of the transfer, I intend to map out the theory for general mapping spaces, not just loop spaces (that’ll give me a bit more motivation to do the transfer since just cut-and-paste is boring!). I’ve just copied over the contents and the introduction so far and haven’t edited them as yet. In particular, although I’ve wikilinked all the original section names, these will get changed as they currently focus on loop spaces.

      The introduction to the original document ended as follows (not copied over to the new version):

      This document began life as notes from talks given at NTNU and at Sheffield so I would like to thank the topologists at those institutions, and in particular Nils Baas, for letting me talk about my favourite mathematical subject. I would also like to thank Ralph Cohen and the “loop group” at Stanford.

      This is by no means a finished document, as an example it is somewhat sparse on references. Any comments, suggestions, and constructive criticism will be welcomed.

      The second paragraph is sort-of stating the obvious as it holds to some extent for any nLab page! And I would love to be able to add some more names to the list in the first paragraph. Again, I hope it goes without saying but I’ll say it anyway: although I anticipate being the main contributor to these pages, it is not my project! I would love it if people read it, add comments, add other stuff, write (constructive) graffiti, link it to other stuff.

    • The entry cover was in a pitiful state. I tried to brush it up a bit. But I am afraid I am still not doing it justice. But also I don’t quite have the leisure for a good exposition right now. What I really want is to create an entry good cover in a moment…

    • stub for Sullivan construction (I got annoyed that the entry didn’t exist, but also don’t feel like doing it justice right now)

    • Because I want to point to it in a reply to the current discussion on the Category Theory Mailing list, I tried to brush up the entry k-tuply monoidal n-category a bit.

      In particular I

      • expanded the Idea section and added some statements that had been missing there;

      • reacted to the old query box discussion there and moved the query box to the very bottom;

      • added a section on k-tuply monoidal ∞\infty-groupoids and ∞\infty-stacks here.

      • added a section on k-tuply monoidal (n,1)(n,1)-categories here

    • I had started an article on AT category (which I originally mis-titled as “AT categories” – thank you Toby for fixing this!), but getting a little stuck here and there. I’m using the exchange between Freyd and Pratt on the categories mailing list (what else is there?) as my reference, but as is so often the case, Freyd’s discussion is a little too snappy and terse for me to follow it down to all the nitty-gritty details.

      There’s a minor point I’m having trouble verifying: that coproducts are disjoint (as a consequence of the AT axioms that Freyd had enunciated thus far where he made that claim, in his main post), particularly that the coprojections are monic. Presumably this isn’t too hard and I’m just being dense. A slightly less than minor point: I’m having trouble verifying Ab-enrichment of the category of type A objects. I believe Freyd as abelian-categories-guru implicitly – I don’t doubt him. Can anyone help?

    • Todd is helping me understand opposite categories beginning with FinSet opFinSet^{op} here.

      This discussion helped prompt some improvement of the page opposite category. When I look at that page now, I see the statement:

      The idea of noncommutative geometry is essentially to define a category of spaces as the opposite category of a category of algebras.

      This reminded me of a remark I made in the “Forward” to a paper I wrote back in 2002, so I’ve now itexified that “Foreward” here:

      Noncommutative Geometry and Stochastic Calculus

      By the way, this also suggests that the category FinSetFinSet is the category of spaces opposite to the category of finite Boolean algebras in the sense of space and quantity.

    • I noticed that recently Konrad Waldorf created a very nice article

      I went through it and added definition/theorem/proof-environments and lots of hyperlinks. Some of them are unsaturated. Maybe somebody feels inspired to create corresponding entries.

    • added the cosimplicial version of the statement to Eilenberg-Zilber theorem and included a reference that gives a proof

    • Eric wanted to know about closed functors, so we started a page. Probably somebody has written about these before, so references would be nice, if anybody knows them. (Google gives some hits that look promising, but I can’t read them now.)

    • I’ve done a bit of housekeeping at Froelicher space. I’ve split the page into pieces, putting each major section into its own section.

      (This will necessitate a little reference chasing at manifolds of mapping spaces, and I need to put in some redirects)

      I’ve put in a definition of curvaceous compactness at topological notions of Frölicher spaces. It works, but I’m not sure if it’s the right one yet.

    • It seems to me that despite so lenghty discussions and entry related to the mapping space-hm adjunction, only the ideal situations are treated (convenient categories of spaces). For this reason, I have created a new entry exponential law for spaces containing the conditions usually used in the category of ALL topological spaces, as well as few remarks about the pointed spaces.

    • Taking the advice that if I write something on the internet, it should be stuck on the n-Lab, I've converted my recent comments in the n-category cafe and some old blog posts into a new page on the relationship between categorification and groupoidification: categorification via groupoid schemes

    • Split off the mapping spaces stuff from local addition into manifolds of mapping spaces. Still plenty to do and things to check (particularly on the linear stuff, and particularly figuring out what “compact” means). Haven’t actually deleted the relevant bit from local addition yet. Also, haven’t put a table of contents at manifolds of mapping spaces since I’ve learnt that the best way to get Urs to read something is to not put a toc in.

    • Casson invariant count SU(2) local systems of integral homology spheres. Thomas considered its holomorphic counterpart which is ultimately related to counting BPS states on Calabi-Yau 3-folds.

      P.S. Hmm. Tags. New option. Great. Is there a list of tags ?

    • I am in a small wave of activity along one of my principal lonegr goals in nlab: the connection between the operator theory and geometry. This is of extreme importance for physics if we ever want to go beyond the TQFTs in quantization program. As Tom Leinster has in his work seen, there are heat-kernel like expansions involved all around the place even when one takes categorical approach and the first terms are of topological nature. This is exactly so in the WKB-expansions where the zeroth term is often the exact value for topological or more general integrable models. Witten's calculation of Witten's index (related to tmf) is an example where such WKB aprpoximations are evaluated and in presence of supersymmetry there are no other terms. Thus I believe that the kantization preferred in nlab is limited to work exactly in simiklar cases and that in general we will have more terms of WKB-like nature involved. We need to develop a categorified WKB method which will unify both.

      On the other hand, the WKB method is not just expansion like in quantum mechanics books, it does involve cocycles right away in usual symplectic geometry. There is so-called Maslov index related to the multidimensional WKB method which has been pioneered by V. Maslov. The quantity which is slowly changing is an analogue of the eikonal in geometric optics, so the whole thing is a generalization of the geometric optics approximation. One can see some aspects of that on (free online, on the AMS web site, under books, here) book on symplectic geometry by Guillemin and Sternberg.

      Harmonic side of the stationary phase approximation (which is just a variant of WKB in fact) is studied for long under the name oscillatory integrals. This is studied especially by Lars Hörmander and the Japanese school of microlocal analysis (btw, that one is the number 3000 entry in nlab!); the differential equations are often decribed via D-modules and in nonlinear case D-schemes which Gorthendieck described via crystals.

      Strangely enough Kashiwara who worked much in microlocal analysis and D-modules has created a notion of crystal bases and crystals of quantum groups but these are NOT related to crystals. Thus I created crystal basis to fix the opinions in the nlab before they go astray...

      I created entry hyperfunction, one of the tools of microanalysis, by Japanese school, a neat version of generalized functions, more flexible than distributions of Laurent Schwarz. They are obtained as boundary values of holomorphic functions (reminds me of BV formalism :)).

    • I added a bit in the functionals section of locally convex space about coordinate projections being continuous for LCTVSs, and that there are counterexamples to this fact without local convexity. This was from memory, I hope I got it right.

      I hope it’s not a fluke that I can edit from home tonight.

      I also hyperlinked my front page of my web a bit, as Urs does (like it’s going out of fashion :), so I can present our model to my company, as I (and some others to whom I have explained it) would like to implement the ’open lab book’ research model we have here. I would loove to be able to do it in instiki (by which I mean the technically minded people), but we may be stuck with an awful free wiki platform, chosen for its ’minimal advertising’ (and I quote!).

      Anyway, as a result, there are a bunch of new stubby pages there, that probably aren’t worth looking at yet.

    • expanded the long-time stub entry (n,1)-topos a little more. Made Mike’s former query box an Example-section.

    • 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.

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