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    • I have tried to make the page torsion look more like a disambiguation page and less like a mess. But only partially successful.

    • I have split off from smooth infinity-groupoid – structures the section on concrete objects, creating a new entry concrete smooth infinity-groupoid.

      Right now there is

      • a proof that 0-truncated concrete smooth \infty-groupoids are equivalent to diffeological spaces;

      • and an argument that 1-truncated concrete smooth \infty-groupoids are equivalent to “diffeological groupoids”: groupoids internal to diffeological spaces.

      That last one may require some polishing.

      I am still not exactly sure where this is headed, in that: what the deep theorems about these objects should be. For the moment the statement just is: there is a way to say “diffeological groupoid” using just very ygeneral nonsense.

      But I am experimenting on this subject with Dave Carchedi and I’ll play around in the entry to see what happens.

    • I have introduced a new section in nlab intitled functorial analysis.

      It talks about the functor of point approach to functional analysis, using partially defined functionals.
    • since the link was requested somewhere, I have created a stub for n-topos

    • In convenient category of topological spaces, I rewrote a little under the section on counterexamples, and I added a number of examples and references. Some of this came about through a useful exchange with Alex Simpson at MO, here.

    • I got a question by email about the equivariant tubular neighbourhoods in loop spaces (as opposed to those defined using propagating flows so I figured it was time to nLabify that section of differential topology of mapping spaces. Of course, in so doing I figured out a generalisation: given a fibre bundle EBE \to B, everything compact, we consider smooth maps EME \to M which are constant on fibres. This is a submanifold of the space of all smooth maps EME \to M. Assuming we can put a suitable measure on the fibres of EE, then we can define a tubular neighbourhood of this submanifold.

      Details at equivariant tubular neighbourhoods. Title may be a bit off now, but it’s that because the original case was for the fibre bundle S 1S 1S^1 \to S^1 with fibre n\mathbb{Z}_n.

      This entry is also notable because I produced it using a whole new LaTeX-to-iTeX converter. Details on the relevant thread.

    • I added a reference to a paper of Connes and Rovelli (1994) and a link (in modular theory) to

      where André Henriques asks about some Connes philosophy. But André quotes in explaining the background to his question, that in full generality there is a homomorphism from imaginary line into the 2-group of invertible bimodules of the given von Neumann algebra MM, which in the presence of state lifts to the homomorphism into Aut(M)Aut(M). I learned just the case when there is a state, and am delighted to hear that this is just a strengthening of some categorical structure which exists even more generally. If somebody is familiar or can dig more on that general case, it would be nice to have such categorical picture in the nnLab entry modular theory.

    • you may recall (okay, probably not ;-) what I once wrote in the entry on exterior differential systems: while in the classical literature these are thought of as dg-ideals in a de Rham complex, we should think of them as sub-Lie algebroids of tangent Lie algebroids.

      Since exterior differential systems over X encode and are encoded by partial differential equations on functions on X, this means that such sub-Lie algebroids are partial differential equations.

      This perspective is amplified much more in the literature on D-modules: I think we can think of a D-scheme as an infinite-order analog of a Lie algebroid, which is the corresponding first-order notion. The Jet-bundle with its D-scheme structure is the infinite-order analog of the tangent Lie algebroid.

      And sub-D-schemes of Jet-D-schemes are partial differential equations, this is what everyone on D-geometry tells you first.

      So I think there is a nice story here.

    • I have updated the reference section on BV formalism by the following:

      i think the Beilinson-Drinfeld book does not treat the classical BV formalism in full generality, even if
      they give a natural language to formalize this (pseudo-tensor, i.e., local operations).

      I changed the corresponding references by saying they give a formalism for quantum BV on algebraic curves.
      The general quantum BV formalism is being studied by Costello-Gwilliam and the formalism of chiral algebras
      in higher dimension that has to be used to generalize Beilinson-Drinfeld to higher dimension is being studied
      by Gaitsgory-Francis in their Chiral Koszul duality article (using infinity categorical localizations to replace model category
      tools for homotopy theory, that are not directly available).

      I also precised the reference to my article about this that uses the language of Beilinson-Drinfeld book and particularly
      local operations, to deal with classical BV formalism for general gauge theories. Beilinson-Drinfeld only treat the
      classical BRST formalism and not classical BV i think (at least not for general base manifold, only for curves).
    • New entry affiliated operator of a C *C^\ast-algebra aka affiliated element. This is important for the circle of entries on algebraic QFT, as the operator algebras are formed by bounded operators, while we typically need unbounded operators like derivative operator to do quantum mechanics.

      I sent a version of that entry but the nnLab stuck in the middle of the operation so I am not sure if I succeeded. So here is the copy:


      Most of the applications of operator algebras stuck in the problem that (hermitean or not) unbounded operators do not form an algebra under composition (or under Jordan multiplication); while the algebras of bounded operators are insufficient as most of applications involve also unbounded operators like the partial derivative operator on L 2( n)L^2(\mathbb{R}^n) which is proportional to the momentum operator in quantum mechanics.


      The motivational problem is typically resolved by considering an operator algebra which contains operators which properly approximate the unbounded operators as close as one wishes, and formalize this by defining the larger class of “approximable” operators by means of operator algebra itself. One way to do this is to define the affiliated elements of C *C^\ast-algebra, or the operators affiliated with the C *C^\ast-algebra. The idea is that if there is an unbounded self-adjoint operator then we can consider its spectral projections; they are bounded and if we include them into the algebra, the convergence of the spectral decomposition will supply the approximation.


      • S. L. Woronowicz, K. Napiórkowski, Operator theory in C *C^\ast-framework, Reports on Mathematical Physics 31, Issue 3 (1992), 353-371, doi, pdf
      • S. L. Woronowicz, C *C^\ast-algebras generated by unbounded elements, pdf
      • wikipedia affiliated operator
    • I was forced to split off the section on infinitesimal cohesion from the entry cohesive (infinity,1)-topos – because after I had expanded it a little more, the nLab server was completely refusing to safe the entry (instead of just being absurdly slow with doing so). I guessed that it is was its length that caused the software to choke on it, and it seems I was right. The split-off subsection is now here:

      cohesive (infinity,1)-topos – infinitesimal cohesion

      Things I have edited:

      • added a bried Idea-paragraph at the beginning;

      • changed the terminology from “\infty-Lie algebroid” to “formally cohesive infinity-groupoid” , making the former a special case (first order) of the latter;

      • expanded the definition of formal smoothness, added remarks on formal unramifiedness in the \infty-context.

    • I wanted to test something in the Sandbox (for this question of David Roberts on the TeX Stackexchange) and it was looking a bit cluttered so I gave it a clean-out.

    • I am about to create D-scheme, but currently the Lab is down and the server does not react to my login attempts…

    • I am about to write something at jet bundle and elsewhere about the general abstract perspective.

      In chapter 2 of Beilinson-Drinfeld’s Chiral Algebras they have the nice characterization of the Jet bundle functor as the right adjoint to the forgetful functor F:Scheme 𝒟(X)Scheme(X)F : Scheme_{\mathcal{D}}(X) \to Scheme(X) from D-schemes over XX to just schemes over XX.

      Now, since D-modules on XX are quasicoherent modules on the de Rham space Π inf(X)\Pi_{inf}(X), I guess we can identify

      Scheme 𝒟(X) Scheme_{\mathcal{D}}(X)


      Schemes/Π inf(X) Schemes/\Pi_{inf}(X)

      and hence the forgetful functor above is the pullback functor

      F(E) E X Π inf(X) \array{ F(E) &\to& E \\ \downarrow && \downarrow \\ X &\to& \Pi_{inf}(X) }

      aling the lower canonical morphism (“constant infinitesimal path inclusion”).

      This would mean that we have the following nice general abstract characterization of jet bundles:

      let H\mathbf{H} be a cohesive (infinity,1)-topos equipped with infinitesimal cohesion HH th\mathbf{H} \hookrightarrow \mathbf{H}_{th}. For any XXX \in \mathbf{X} we then have the canonical morphism i:XΠ inf(X)i : X \to \mathbf{\Pi}_{inf}(X).

      The Jet bundle functor is then simply the corresponding base change geometric morphism

      Jet:=(i *i *):H/XH/Π(X) Jet := (i^* \dashv i_*) : \mathbf{H}/X \to \mathbf{H}/\mathbf{\Pi}(X)

      or rather, if we forget the 𝒟\mathcal{D}-module structure on the coherent sheaves on the jet bundle, it is the comonad i *i *i^* i_* induced by that.

      Does that way of saying it ring a bell with anyone?

    • started a Reference entry FHT theorem with a brief rough statement of what the theorem says. For the moment mainly in order to include pointers to where in the three articles the theorem is actually hidden (I think it is hidden quite well… ;-)

    • I am hereby moving the following discussion from information geometry to here:

      Tim Porter: I have looked briefly at the Methods of Info Geom book and it seemed to me to be distantly related to what the eminent statistician David Kendall used to do. He and some coauthors wrote a very nice book called: Shape and Shape Theory (nothing to do with Borsuk’s Shape Theory). The theory may be of relevance as it used differential geometric techniques. (Incidently there are some nice questions concerning the space of configurations of various types that would be a good source for student project work in it.)

      My query is whether the link is a strong one between the Amari stuff and those Kendall Shape space calculations. Kendall’s theory and some similar work by Bookstein is widely used in identifcation algorithms using a feature space. In case the link is only faint I will leave it at that for the moment. Any thoughts anyone?

      Eric: I wrote some stuff here, which is now relegated to Revision 5. I’ve rewritten most of the material here.

    • On the basis of wikilinking everything, I discovered that orthogonal structure didn’t exist, so I created it. Being me, I gave it my infinite dimensional slant. Clearly there’s lots that could be said here, so it’s a middling stub.

    • First stab at propagating flows (highly tempted to put in a redirect for propogating flows). I wrote it without reference to either my article or Veroniques’ in the hope that by being forced to look at it afresh, I’d get the argument right. I’m not convinced that I managed it so I’ll need to polish it considerably.

    • I have added reference pointers to Moritz Groth’s document on “Derivators, pointed derivators and stable derivators” to the relevant entries, such as stable derivator.

      Mike, I forget if you mentioned that before or not. I only learned of his work today. Part of his PhD thesis with Schwede.

    • I badly need to polish the nnLab entries related to path integrals. Today a student asked me how the pull-push operation in string topology is a remnant of a quantum path integral. So a started writing now

      So far there is the description of the archetypical path integral for the quantum particle propagating on the line in terms of pull-tensor-push.

    • I moving the following old discussion from dg-algebra to here:


      A previous version of this entry gave rise to the following discussion

      +–{.query} Zoran, why would you not say that this is ’following the product rule from ordinary calculus’, as I wrote? Not that this can be proved like the product rule can, but it's an easy mnemonic (and a similar one works for direct sums too). —Toby

      I find it very confusing for me at least. The Leibniz rule is about the coproduct in a single algebra; here one has several algebras with different differentials, not a single derivative operators, and not acting on a tensor square of a single algebra, so it is a bit far. If A=BA=B then I would be happy, but otherwise it is too general. —Zoran

      You mean that if A=BA = B, then the Leibniz rule is a special case of this? Then surely it is also a special case of the more general case without A=BA = B? Anyway, I think that it's more an example of categorification than generalisation. —Toby

      For some special algebras this is true. For example, the dual of symmetric algebra as a Hopf algebra can be identified with the infinite order formal differential operators with constant coefficients (the isomorphism is given by evaluation at zero). Thus the Leibniz rule for derivatives is indeed the dual coproduct to the product on the symmetric algebras. There are braided etc. generalizations to this, and a version for computing the coproduct on a dual of enveloping algebras. In physics the addition of momenta and angular momenta for multiparticle systems is exactly coming from this kind of coproduct. But in all these cases the operators whose product you are taking live in a representation of a single algebra. — Zoran


    • A coupld additions to measurable space that I've been sitting on for a while, and which I've realised that I'm not going to write more clearly anytime soon.

      But someday I would like to move a lot of the discussion about various approaches to measure theory and make measurable space itself simpler, with pointers to variations.

    • I have removed this sentence from AnonymousCoward:

      (Well, usually. Urs Schreiber —or for all we know, possibly somebody impersonating him (^_^)— has managed to keep his IP address out sometimes.)

      This makes it sound as if I did something intentionally to hide my IP, which is not the case. Rather there must be a problem with the software, if something that should not have happened did happen.

      I have also removed the following old discussion, which is better had here on the forum:

      Eric: Can we change this? I am not anonymous, but I also do not want my IP listed (since it resolves to my employer, which I think should be private.) I guess I can always just not post from work, but small distractions now and then are nice.

      Toby: IP addresses are almost always logged by web software, even for readers; in the past, these logs were usually deleted after a while, but now storage space is so cheap that this may no longer be true. People like to have the IP address available in case of problems —spam, DoS attacks, etc—. I like having that sort of information publicly available, rather than tucked into logs that are hidden behind passwords, to prevent the devlopment of hierarchies.

      But if you want to be anonymous on the web, try searching for ’web proxy’ or the like. However, Jacques's software makes a fair attempt to defeat these, since they are often used to spam. (Even in general, I don't know how well they work, and ultimately they become the people with the secret information.)

      Toby: I see that Urs managed to post from ’from bogus address’ today (June 27). Maybe we should ask him what he did differently!

      Eric: I don’t mind if administrators can see my IP for security reasons, but it is not clear what purpose it serves to actually display it publicly for all to see. For example, I can see the IP addresses of people who comment on my blog, but it is not displayed for everyone to see.

      Toby: That creates a hierarchy (of information if not power, but one leads to the other) where administrators are above everybody else. The wiki way gives the same information to everybody.

    • New stub copyright both about copyright attitude of the nnCommunity and as a place to collect links to interesting analysis of copyright, free literature, protection from plagiarism and similar issues. It also links to citations (zoranskoda).

    • I created a page emptypage. It would belong to meta category of pages but I do not want to attach even that label to it. I want it empty, I want it orphane, non-aliased and non-classified, truly minimal content and minimal sourcecode page.

      With one click of the mouse I call the label of nlab:HomePage in my bookmarks bar, and then I change the URL by hand or go from HomePage to one of the links or use the search from there. If I am on slow connection, sometimes even HomePage loads longer. I think that some other users can smartly use the initial page like that. HomePage has information for newcomers, experienced users can sometimes prefer emptypage as their cleaner and leaner nnLab homepage.

      So emptypage is a quick way to see that the lab is up with a minimal length page and to get the basis for nnLab search window or to change the URL without the cost of the HomePage load and HomePage html display time. Now with HomePage having also an additional Terms of usage section it grown today another bit more, so a reason more to create emptypage and to hopefully leave it empty.

      I use emptypage to have it easier to type than empty page.

      I hope other people won’t find it offending that I created a lean-expert-user depart point without consulting others, but I think it has obvious usages for some and it is not on the way to others, I hope.

    • I’ve been thinking a lot about degeneration of Hodge to de Rham spectral sequence lately. I checked out the page on the nlab about it. I saw that there was a link to Cartier operator but no page, so I created it.

      This actually got me thinking. In some sense degeneration at E 1E_1 is “intrinsic” to the derived category D(X)D(X) (I just made that up based on what I wrote in the article). There is a naive way to try to prove that if XX and YY are derived equivalent and if the SS degenerates for one, the other should too. I couldn’t see a way to make it work. Is there an obvious reason this should be true, or an obvious counterexample?

    • An anonymous correspondent has put a question on lax functor, or rather has edited a previous query.