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    • added to Hilbert bimodule a pointer to the Buss-Zhu-Meyer article on their tensor products and induced 2-category structure.

    • I have now created relative category.

      Question: Does the transferred model structure on RelCat\mathbf{RelCat} resolve Rezk’s [2001] conjecture that the classification diagram of a model category is weakly equivalent to its simplicial localisation? The N ξN_\xi functor looks very close to computing the hammock localisation to me…

    • I have added to groupoid convolution algebra the beginning of an Examples-section titled Higher groupoid convolution algebras and n-vector spaces/n-modules.

      Conservatively, you can regard this indeed as just some examples of applications of the groupoid convolution algebra construction. But the way it is presented is supposed to be suggestive of a “higher C*-algebra” version of convolution algebras of higher Lie groupoids.

      I have labelled it as “under construction” to reflect the fact that this latter aspect is a bit experimental for the moment.

      The basic idea is that to the extent that we do have groupoid convolution as a (2,1)-functor

      C:GrpdAlg b op2Mod C \colon Grpd \to Alg_{b}^{op} \simeq 2Mod

      (as do do for discrete geometry and conjecturally do for smooth geometry), then this immediately means that it sends double groupoids to convolution sesquialgebras, hence to 3-modules with basis (3-vector spaces).

      As the simplest but instructive example of this I have spelled out how the ordinary dual(commutative and non-co-commutative) Hopf algebra of a finite group arises this way as the “horizontally constant” double groupoid incarnation of BG\mathbf{B}G, while the convolution algebra of GG is the algebra of the “vertically discrete” double groupoid incarnation of BG\mathbf{B}G.

      But next, if we simply replace the bare Alg b op2ModAlg_b^{op} \simeq 2 Mod with the 2-category C *Alg bC^\ast Alg_b of C *C^\ast-algebras and Hilbert bimodules between them and assume (as seems to be the case) that C *C^\ast-algebraic groupoid convolution is a 2-functor

      LieGrpd C *Alg n op LieGrpd_{\simeq} \to C^\ast Alg_n^{op}

      then the same argument goes through as before and yields convolution “C *C^\ast-2-algebras” that look like Hopf-C*-algebras. Etc. Seems to go in the right direction…

    • seeing the announcement of that diffiety summer school made me think that we should have a dedicated entry titled cohomological integration which points to the aspects of this discussed already elswhere on the nLab, and which eventually lists dedicated references, if any. So I created a stub.

      Does anyone know if there is a published reference to go with the relevant diffiety-school page ?

    • dropped some lines into a new Properties-section in the old and neglected entry bibundle. But not for public consumption yet.

    • I felt we were lacking an entry closure operator. I have started one, but don’t have more time now. It’s left in a somewhat sad incomplete state for the moment.

    • I could have sworn that we already had entries like “topological ring”, “topological algebra” or the like. But maybe we don’t, or maybe I am looking for the wrong variant titles.

      I ended up creating a stub for topological algebra now…

    • I have added to C-star algebra the statement that the image of a C *C^\ast-algebra under an *\ast-homomorphism is again C *C^\ast.

      Also reorganized the Properties-section a bit and added more references.

    • the entry groupoid could do with some beautifying.

      I have added the following introductory reference:

      • Alan Weinstein, Groupoids: Unifying Internal and External Symmetry – A Tour through some Examples, Notices of the AMS volume 43, Number 7 (pdf)
    • I have started adding some references to

      on modules (C *C^\ast-modules) of (continuous, etc..) convolution algebras of topological/Lie groupoids.

      I still need to look into this more closely. A motivating question for this kind of thing is:

      what’s the right fine-tuning of the definition of modules over twisted Lie groupoid convolution algebras such that for centrally extended Lie groupoids it becomes equivalent to the corresponding gerbe modules?

      This seems fairly straightforward, but there are is some technical fine-tuning to deal with. I was hoping this is already stated cleanly in the literature somewhere. But maybe it is not. Or maybe I just haven’t seen it yet.

    • Wrote a quick note at centrally extended groupoid and interlinked a little, for the moment just motivated by having the link point somwhere.

    • as mentioned in another thread, I have expanded the Idea-section at polarization in order to highlight the relation to canonical momenta (which I also edited accordingly).

    • felt like making a terminological note on phase and phase space in physics (and linked to it from the relevant entries).

      If anyone has more information on the historical origin of the term “phase space”, please let me know.

    • started a dismabiguation page for phase. Feel invited to add further meanings.

    • Just in case you see me editing in the Recently Revised list and are wondering:

      I have created and have started to fill some content into semiclassical state. But I am not done yet and the entry is not in good shape yet. So don’t look at yet it unless in a mood for fiddling and editing.

    • I started an entry classical-to-quantum notions - table for inclusion in “Related concepts”-sections in the relevant entries.

      This is meant to clean up the existing such “Related concepts”-lists. But I am not done yet with the cleaning-up…

    • New entry semiclassical approximation. It requires a careful choice of references. The ones at the wikipedia article are catastrophically particular, 1-dimensional, old and non-geometric and hide the story more than reveal. Stub Maslov index containing the main references for Maslov index.

    • I created Galois topos following Dubuc’s article.

      But I must be missing something about the notation: does it really mean to say that AA is an ΔAut(A)\Delta Aut(A)-torsor, as opposed to saying that it is associated to an ΔAut(A)\Delta Aut(A)-torsor?

    • There have been two empty pages created lately, both anonymous. They are at Riemann sphere and quasi inverse. It looks as if both were attempts to add something that was aborted.

    • This page, wall crossing in Aarhus, refers (in the future tense) to a course in 2010. The webpage link is broken as well. Does anyone have a link that could replace that one?

    • Thought I would flag up that there have been two of these lately, ideal in semigroups and liars paradox. I waited to see if their ‘authors’ were going to come back and correct them, but so far they have not.

      The entry Mochizuki's proof of abc is non-standard in form but has been updated by someone called Daniel.

    • created Poisson tensor just for completeness, to be able to point to it from related entries.

    • Added a little to the Idea-section of holonomy groupoid. But this deserves to be further expanded upon.

    • Mentions of the category SetSet occur all over the nLab, but with quite a bit of plasticity of meaning. I thought it might be good to have another look at the entry Set and try to describe this plasticity as considered along various axes, to help readers who might be puzzled by “just what does the nLab think the category of sets is?” For example, one reads that the category of sets has marvelous properties such as being a well-pointed topos, and then a little further down one sees that SetSet is not a topos according to predicative mathematics. This could be very confusing. Similarly, there are some pages in the nLab that assume SetSet satisfies AC without batting an eye, while others discuss arcane weaker choice principles that SetSet might satisfy. I think we need to be a just a bit more up-front about this, right on the page Set.

      In the definition section on Set, I made a meager start on this by declaring that the nLab adopts a ’pluralist’ position on the matter of sets and SetSet, and jotted down a few of the possible axes (“axises”, if I were James Dolan) of meaning and interpretation that guide how one thinks of SetSet, e.g., predicative vs. impredicative, classical vs. intuitionist, selection of choice principles, and others. I didn’t think really hard about this, but it might suggest useful ways of organizing the page.

      I left out other axes such as “structural vs. material”, and said nothing about type theory. The page set talked more about this; I envision Set as concentrating more on properties of the category of sets.

      I got to thinking about this when I began to wonder how Toby thinks about SetSet, which is maybe different from how I usually think about it. (Usually it feels slightly alien to me to posit say WISC as a possible choice principle for the category of sets, which for me usually connotes a model of ETCS – normally I’d think of WISC instead as a possible axiom for a topos or a pretopos.) I was wondering whether Toby had a kind of “bottom line” for SetSet, say for example “SetSet for me means at least a well-pointed topos with NNO, unless I choose to adopt a predicative mode”, or something like that. Anyway, discussion is invited.

    • After a few days’ editing, I’m announcing absolute differential form (using my neologism). This is a notion of differential form that can be integrated on a completely unoriented submanifold. Examples from classical differential geometry include the arclength element on a Riemannian manifold and |dz|{|\mathrm{d}z|} on the complex plane.

      Since there are classical examples, people must have thought about these before me, but I have never heard of them. Absolute differential forms are not linear (although they must satisfy a restricted linearity condition), and many typical examples are not smooth (although they are still continuous), so they don’t show up in the usual classification theorems. Has anybody heard of them before?

    • created brief remark at Kostant-Souriau extension

      (beware the hyphen bug, sometimes only Kostant Souriau extension will work, not sure why and when)

      (the hyphen bug combined with the cache bug combined with the low responsiveness make for a special experience…)

    • New stubs N-complex (the homological algebra where d N=0d^N = 0 with N>2N\gt 2) and Michel Dubois-Violette. This interested me somewhat over a decade ago. Unfortunately, I missed the seminar talk yesterday in Zagreb by one of my colleagues, Pavle Pandžić, who found with his collaborator, very recently, that a more general and more insightful redefinition of Dirac cohomology, suggested by concrete applications in representation theory, involves the homological algebra of NN-complexes. I hope there will be some writeup soon available.

    • I made linearly independent subset to satisfy a link. I put in something about the free-forgetful adjunction and something else about constructive mathematics.

    • expanded a bit the discussion of morphisms of sites at site

    • There was a parity error at n-group; I fixed that and put in the low-dimensional examples.

    • I have created a diambiguation entry at Artin-Mazur codiagonal, as the old links at bisimplicial set lead to the entry on the codiagonal of a coproduct. I have used total simplicial set as the preferred term. Perhaps a more detailed discussion of this might be useful, but I have not got the time at the moment. (I am very slow at doing diagrams, :-( )

    • Wrote generic proof with some comments about a couple seemingly weaker versions of the axiom of choice that I've never seen mentioned anywhere before (has anyone else?). Toby and I noticed these a little bit ago while thinking about exact completions, but I just now realized that they're actually good for something: proving that the category of anafunctors between two small categories is essentially small (in the "projective" way).

    • I wrote eventuality filter, although maybe this was unnecessary, and as it was mostly already there at net. Then I took some of the logic from there and adapted it to null set.

    • created free field theory with the formalization in terms of BV-complexes by Costello-Gwilliam.

    • Does anyone know if we have a discussion, somewhere, of the theorem of Thomason linking homotopy colimits with Grothendieck constructions. I have looked in places that I thought were likely but found no trace of it, but sometimes things get buried in entries on other topics so are difficult to find.

    • New entry critics of string theory to collect the references on controversies. I think they are often rambling and vague, not technically useful s the main references we want to collect under string theory and books in string theory. I have changed the sentence in string theory about mathematical definition of parts to somewhat more precise

      But every now and then some aspect of string theory, some mathematical gadget or consequence found there is isolated and redefined independently and mathematically rigorously, retaining many features originally predicted.

      The point is that most often one does not make rigorous the way some thing is defined via string theory, but one isolates an invariant of manifolds for example and defines a similar one via completely different foundations. The typical example is quantum cohomology which is defined in geometric terms and not in terms of field theory any more.

      I have one disagreement with the entry: it says that topological quantum field theory has been discovered as part of string theory research, This is not true, TQFTs were found in 1977, 1978, 1980 articles of Albert Schwartz which had nothing to do with string theory. Only much later Atiyah’s formulation is influenced by string theory.

    • I finally gave The convenient setting of global analysis a category: reference-entry. Started adding pointers to it from the References-section of some relevant entries. But there will be many more left.

    • I moved some discussion from bicategory to weak enrichment, a new page. (Possibly it was already moved somewhere else, since Mike had already deleted it, but I couldn't find it.)