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    • Urs, while it is good that spectral theorem is included into functional analysis table of contents, and it has functional analysis toc bar, I do not like that spectral theory is also included and also has this toc bar. My understanding is that spectral theory is much wider subject on the relation between the possibly categorified and possibly noncommutative function spaces (sheaf categories, noncommutative analogues) and the specifical “singular” features of those like prime ideals, like certain special objects in abelian categories, points of spectra in operator framework etc. In any case, in nnPOV, it is NOT a part of functional analysis, though some manifestations are. Like the concept of a space is not a subject of functional analysis, though some spaces are defined in the language of operator algebras. I find spectral theory on equal footing like space, “quantity” etc. Of course, the entry currently does not reflect this much (though it has a section on spectra in algebraic geometry), but it eventually will! Thus I will remove it from functional analysis contents.

      One should also point out that using generators in the proof of Giraud’s reconstruction theorem of a site out of a topos is a variant of spectral idea: like points form certain spaces, so the generators of various kind generate or form a category. This is behind many spectral constructions (including recent Orlov’s spectrum which is very laconic but stems from that) and reconstruction theorems and if the category corresponds to coherent sheaves over a variety than often the geometric features of the variety give certain contributions to the spectrum.

    • Can someone look at Three Roles of Quantum Field Theory. There was an unsigned change there and a box that does not work. I do not know what was intended so will not try to fix it.

    • New entry Dmytro Shklyarov; he seems to be now in Augusburg. Lots of interesting recent work in several subfields of our interest. I did not know where to put his 2-representations paper into 2-vector space as the bibliography is scattered there with some classification of subtopics.

    • At Urs’ urging, I have created functional analysis - contents. It needs considerable extending; and I’ve yet to include it anywhere.

      As hinted by the contents, I plan to move the diagram from TVS to its own page (but still include it on TVS).

    • I wrote about these at measurable space, following to reference to M.O answers by Dmitri Pavlov that were already being cited.

    • have a look into (the) future

    • I have added some new material to Boolean algebra and to ultrafilter. In the former, I coined the term ’unbiased Boolean algebra’ for the notion which describes Boolean algebras as equivalent to finite-product-preserving functors Fin +→SetFin_+ \to Set from the category of finite nonempty sets, and the term kk-biased Boolean algebra to refer to the multiplicity of ways in which Boolean algebras could be considered monadic over SetSet.

      In ultrafilter, I added some material which gives a number of universal descriptions of the ultrafilter monad. This is in part inspired by some discussions I’m having with Tom Leinster, who remarked recently at the categories list that the ultrafilter monad could be described as a codensity monad. All this is related to the unbiased Boolean algebras and to the remarks due to Lawvere, which were described on an earlier revision; this material has been reworked.

    • 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 E→BE \to B, everything compact, we consider smooth maps E→ME \to M which are constant on fibres. This is a submanifold of the space of all smooth maps E→ME \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 1→S 1S^1 \to S^1 with fibre ℤ 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:

      Motivation

      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(ℝ n)L^2(\mathbb{R}^n) which is proportional to the momentum operator in quantum mechanics.

      Idea

      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.

      Literature

      • 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)

      with

      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 H↪H th\mathbf{H} \hookrightarrow \mathbf{H}_{th}. For any X∈XX \in \mathbf{X} we then have the canonical morphism i:X→Π 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/X→H/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:

      Discussion

      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

      =–

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