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    • There doesn’t seem to be a discussion for this page differential cohesion and idelic structure. Is this to be the general page for ’inter-geometry’?

      If so, it might be worth recording An Huang, On S-duality and Gauss reciprocity law, (Arxiv).

    • Began stub for Tambara functor. Neil Strickland’s, Tambara Functors, arXiv:1205.2516 seems to be a good reference.

      Seems like it’s very much to do with pullpush through polynomial functors, if you look around p. 23.

      I would try to say what the idea is, but have to dash.

    • In two recent threads [1, 2] I had started to look into elementary formalization of the following obstruction problem in higher geometry:

      given

      • a Klein geometry HGH \to G,

      • a WZW term L WZW:G/HB p+1𝔾 conn\mathbf{L}_{WZW} : G/H \longrightarrow \mathbf{B}^{p+1} \mathbb{G}_{conn};

      • a Cartan geometry XX modeled on G/HG/H

      then:

      • what is the obstruction to globalizing the WZW term to XX?

      Here are first concrete observations, holding in any elementary \infty-topos (meaning: this may be proven using HoTT, not needing simplicial or other infinite diagrams):

      First, a lemma that turns the datum of a global WZW term Fr(X)B p+1𝔾 connFr(X) \longrightarrow \mathbf{B}^{p+1}\mathbb{G}_{conn} on the frame bundle of XX (each of whose fibers looks like the formal disk 𝔻\mathbb{D} around the base point, or any other point, in G/HG/H) into something closer to cohomological data on XX. In the following Fr(X)Fr(X) may be any fiber bundle EE and B p+1𝔾 conn\mathbf{B}^{p+1}\mathbb{G}_{conn} may be any coefficient object AA.

      Lemma. Let EXE \to X be an FF-fiber bundle associated to an Aut(F)Aut(F)-principal bundle PXP \to X. Then AA-valued functions on EE are equivalent to sections of the [F,A][F,A]-fiber bundle canonically associated to PP.

      Proof. By the discussion at infinity-action, the universal [F,A][F,A]-fiber bundle [F,A]/Aut(F)BAut(F)[F,A]/Aut(F)\to \mathbf{B} Aut(F) is simply the function space [F,A] BAut(F)[F,A]_{\mathbf{B} Aut(F)} formed in the slice over BAut(F)\mathbf{B} Aut(F), with FF regarded with its canonical Aut(F)Aut(F)-action and AA regarded with the trivial Aut(F)Aut(F)-action.

      Now, by universality, sections of P×Aut(F)[F,A]XP \underset{Aut(F)}{\times} [F,A] \to X are equivalently diagonal maps in

      [F,A]/Aut(F) X BAut(F) \array{ && [F,A]/Aut(F) \\ & \nearrow & \downarrow \\ X & \longrightarrow & \mathbf{B} Aut(F) }

      But by Cartesian closure in the slice and using the above, these are equivalent to horizontal maps in

      E = P×Aut(F)F A×BAut(F) BAut(F) \array{ E & = & P \underset{Aut(F)}{\times} F && \longrightarrow && A \times \mathbf{B}Aut(F) \\ && & \searrow && \swarrow \\ && && \mathbf{B}Aut(F) }

      Finally by (BAut(F)BAut(F) *)(\underset{\mathbf{B}Aut(F)}{\sum} \dashv \mathbf{B}Aut(F)^\ast) this is equivalent to maps EAE \to A. \Box

      \,

      [ continued in next comment ]

      [1] Cartan geometry, supergravity and branes

      [2] axiomatic tangent structure of etale homotopy types

    • created a bare minimum at harmonic map (for the moment just so as to have a place to record the reference given there)

    • I have created pages on quandles and racks. I recently spent a bunch of time improving the Wikipedia article on this topic due to a comment by Vaughan Pratt. I decided to transfer some of this information to the nLab, in a style somewhat more suitable for professional mathematicians.
    • I added a new section to Bayesian reasoning, Exchangeability, which outlines the de Finetti Representation theorem. As indicated, there’s a multivariate version. This was used to talk about Bose-Einstein statistics.

      I wonder if anything interesting would happen with a HoTT rendition of statistical meachanics.

    • Someone has got the Euler-Lagrange equation page to redirect to hollymolly, and added that word at the bottom. How do we undo such vandalism?

    • during a talk on homotheties of Cartan geometries that I heard yersterday, it occurred to me that this concept has an immediate simple general abstract formulation in differential cohesive homotopy theory. Made a note on this now at homothety.

    • Someone has once again created a ’first slide’ entry! see G.822025. I will blank it to an ’empty’ page (147) (There was also another page withjust four characters on which has become empty 148.)

    • There was an ancient query box discussion sitting in the entry dg-Lie algebra which hereby I am moving from there to here.


      begin of ancient discussion


      +–{: .query}

      Tim: I have changed the wording that Zoran suggested slightly. Of course, a dgla is an internal Lie algebra, a term that needs making precise in an entry, but then we must make precise the tensor product, and the symmetry. All that abstract baggage is, of course, in other entries, but I think it best to avoid the term ’simply’. I have heard it expressed that category theorists tend to use the term ’simply’ aand other similar terms too much from the point of view others working in neighbouring disciplines.

      For instance, if someone knows de Rham theory from a geometric viewpoint, we know that in the long run it will be useful for them to understand the differential graded algebra from a categorical viewpoint as that is one of the most fruitful approaches for geometrically significant generalisations and applications BUT the debutant can get very put off by thinking that they have to understand lots of category theory before they can start understanding the de Rham complex. In fact coming from that direction they can understand the category theory via the de Rham theory. So I suggest that we simply avoid ’simply’!!

      I know some researchers in other subject areas are looking with interest to the nLab as a quick means of entry into some interesting mathematics and a handy reference for definitions and background. That is great but it perhaps means that we have to be a bit careful about our natural feeling that the categorical approach is nearly always the ’best’. ’Simply’ is one problem, another is, I think, use of diagrams rather than formulae. My feeling is that both should be given (though the diagrams are more difficult to get looking nice).

      Urs: these are all good points. In general I believe it will be good to offer different perspectives in an nnLab entry, and explain what they are useful for, each. I take the point that the word “simply” for the categorical perspective may raise unintended feelings, so maybe it should be avoided or at least not left uncommented.

      But we should also not hide the important point here, which is hinted at by the word simply: I think that the important point is that the abstract category-theoretic formulation which packages a long list of detailed definitions in a single statement such as “internal Lie algebra” allows us to recognize that that list of definitions is right.

      There are many definitions that one can dream up. But some are better than others and category theory can explain why.

      For instance I have seen experts who calucalted with differential graded algebra all day long be mystified by why exactly all the sign rules are as they are. The best explanation they had was: it works and yields interesting results. They were positively interested to learn that all these signs follow automatically and consistently by realizing that differential graded algebra is algebra internal to the category of chain complexes.

      This doesn’t mean that it is best to introduce DGCA in this internal language. But it does mean that it is worthwhile pointting out that lots of nitty-gritty details of definitions can “simply” be derived by starting with an abstract internal definition and then turning the crank.

      Tim: I could not have put it better myself. I was wondering if there might not be some way in which this viewpoint might not be expressed explicitly. Perhaps David C has some thoughts.. sort of ’the unreasonable effectiveness of categorical language’?

      My intention for my own contribution (with help hopefully) is to gradually add glosses in the lexicon entries so as to help interpret in both directions, categorically,and geometrically.

      For instance, in the construction of the cobar one take the tensor algebra of the suspension of the cokernel (is it?) of the augmentation. WHY?!!!!!!! How can one understand this? Magic? It works? In fact it is still a bit of a mystery to me and saying that it comes from such and such a categorical property still needs spelling out for me. I have asked rational homotopy theorists and have partially understood things from their point of view but there are still gaps in my understanding of it and some of them worry me!

      Toby: One should be able to say something like, ’From a category-theoretic perpsective, a differential graded Lie algebra is simply an internal Lie algebra in an appropriate category of chain complexes.’. This advertises what Urs says, that definitions come automatically from the category-theoretic perspective, without pretending that this will be simple to anyone coming from outside that perpsective.

      Zoran Škoda: Tim, your question about the intricacies of cobar construction in the category of chain complexes is an interesting one, which I can not fully answer, specially in a short answer. However, still the categorical picture simplifies the viewpoint and the definition at least,and gives a direction how to proceed there as well. Given a dgca C one looks at the functor Tw(C,A) assigning to an algebra A the set of solution of the Maurer-Cartan equation dt+t*t=0d t + t*t = 0 where ** is the convolution product. Cobar construction is the (co)representative of this covariant functor. If you take Tw(C,A) as a contravariant functor on the coalgebras, for fixed A, then its representative is the bar construction (this is said in different words in entry twisting cochain). So bar and cobar construction are simply representatives of very natural functors; accidents of the realization of these functors by formulas in Ch are a bit unfortunate as you pointed out.

      =–


      end of ancient discussion


    • I wanted to add some references on WKB approximation, but found that we have two entries semiclassical approximation and WKB method. WKB or semiclassical expansion is one and the same thing: asymptotic expansion of quantum mechanical amplitudes in Planck constant. On the other hand, “WKB method” is often used to limit considerations just to the stationary phase approximation way of doing the expansion, rather than say to the path integral equivalent (the latter anyway used mainly in physics treatments of semiclassical expansion only).

      1. Historically WKB or WKBJ (J for Jeffreys) method or approximation has been studied only in one dimension till works of Maslov and others in late 1950s, when the multidimensional analogue has been found. The asymptotics of wave type equations has been studied more generally by Maslov, Hoermander and others as the theory of Fourier differential operators where the stationary phase approximation is the main tool. Mathematically, WKB is precisely the stationary phase approximation and it has been used much earlier in optics as so called geometrical optics approximation.

      One can “historically” limit to just one dimension and just to asymptotics of integral expressions in first order, so in some sense one can limit to some particular case as WKB approximation, but for a modern researcher, WKB and semiclassical method is one and the same thing. I can hardly split the discussion and references to the two entries, so I would rather have them merged into one entry and restrict any mention of the difference in scope to a historical subsection. What do you think about it (Urs, especially). (In fact it makes some sense to rename WKB method entry into 1-dimensional WKB method and to discuss just the old early theory there).

      1. I see that the table in semiclassical approximation says that formal deformation is in all orders while semiclassical just in first (or finite? not clear from the table) order. This is not true, semiclassical expansion is sometimes considered to all orders. But it is an expansion of complex valued functions understood as asymptotic series, and summability issues and analysis of rapidly oscillating functions is in the center of attention. Thus formal means formal, in the sense of formal power series. Semiclassical is asypmtotic expansion, not only formal. Nothing to do with first order !

      At semiclassical approximation, I added references on so called exact WKB method, very popular recently, stemming from Voros 1983, where one looks at WKB expansion to all orders and understands it in the sense of Borel summability.

      • A. Voros, The return of the quartic oscillator. The complex WKB method, Annales de l’institut Henri Poincaré A39:3, 211-338 (1983) euclid

      • Alexander Getmanenko, Dmitry Tamarkin, Microlocal properties of sheaves and complex WKB, arxiv/1111.6325

      • Kohei Iwaki, Tomoki Nakanishi, Exact WKB analysis and cluster algebras, J. Phys. A 47 (2014) 474009 arxiv/1401.7094; Exact WKB analysis and cluster algebras II: simple poles, orbifold points, and generalized cluster algebras, arXiv:1409.4641

    • Shelah’s main gap

      sorry, first created it in David Corfields’ web, then in nnLab. DId not keep track where I was.

    • I gave D-brane geometry a minimum of content and references. This is what in string theory was called “D-geometry” in the glory 90s and so I added a disambiguation line at the top of the latter entry.

    • We did not have the page on coalgebraic logic so I just created as a place for links for now, we could expand on it later.

    • I have created random variable with some minimum context.

      In addition I have added pointers to Kolmogorov’s original book and to some modern lecture notes to probability theory and some related entries.

      I have briefly cross-linked probability space with possible worlds, indicating a similarity of concepts and an overlap of implementations.

    • added to differential operator the characterization via bundle maps out of a jet bundle, together with the note that this means that differential operators are equivalently morphisms in the co-Kleisli category of the Jet bundle comonad.

    • I look at ind-cocompletion and pro-completion issues these days. New entry cocompletion. References at many related entries. Notably (at inaccessible cardinal)

      • Andreas Blass, Ioanna M. Dimitriou, Benedikt Löwe, Inaccessible cardinals without the axiom of choice, Fund. Math. 194 (2007) 179-189 pdf

      We consider four notions of strong inaccessibility that are equivalent in ZFC and show that they are not equivalent in ZF.

      (Strange: if I paste ZF and ZFC in font from the Fundamenta page abstract, the nForum truncates everything starting with ZF. This way I lost part of the text which I wrote after this.)

      Note that the wikipedia and some other sources have an outdated link for the Blass et al. paper, at a Dutch site. This pdf link is to the Polish Fundamenta site, and works as of now.

    • In the Idea section of Zariski site, I included a little patch which includes the little site notion, as well as the big site.

    • Prompted by this discussion, I added a minimal explanation of the terminology “name of an object” to the page universe in a topos, right below the first diagram. Please feel free to correct if this is not right.

      Since there are several pages talking about universes I also don’t know if that’s the best place for that edit.

    • I have edited and expanded wall crossing a little

      One question to Zoran:

      you have designed the entry to cover the notion in great generality. But most of the references that you already had, and now also all that I have added, concern wall crossing of BPS states. Eventually we need to do something to make the entry more systematic on this point. Should we split off an “wall crossing of BPS states”, maybe?

    • added to Descartes as section On space, matter and mechanics with some quotes on Descartes’ picture of mechanism.

      (I was looking for sources that would argue clearly that Descartes’ mechanism is closer to modern continuum mechanics than to modern point particle mechanics. I found something, but I imagine there might be better such sources still.)

    • I had thought we had an entry on the van Est isomorphism, but maybe we didn’t. Have started a bare minimum, just so as to have the link.

    • This week I am at a workshop in Bristol titled Applying homotopy type theory to physics, funded by James Ladyman’s “Homotopy Type Theory project”. David Corfield is also here. The program does not seem to be available publically, but among the other speakers that the nnLab community knows is also Jamie Vicary.

      Myself, I will give a survey talk titled “Modern physics formalized in Modal homotopy type theory” (which maybe should rather have “to be formalized” in the title, depending on how formal you take formal to be). I am preparing expanded notes to go with this talk, which I am keeping at

      This is still a bit rough at some points, but that’s how it goes.

      I currently also have a copy of the core of this material in one section at Science of Logic, replacing the puny previous section on formalization that was there. While it’s not puny anymore, now maybe it’s too long and should be split off. But just for the time being I’ll keep it there.

      If you look at it, you’ll recognize a few points that I tried to discuss here lately, more or less successfully. This here is not meant to force more discussion about this – we may all be happier with leaving it as it is – it’s just to announce edits, in case anyone watching the RecentlyRevised charts is wondering.

    • I wrote a subsection at completely distributive lattice on the case of Boolean algebras, showing that they are the same as complete atomic Boolean algebras.

    • A little bit of a page at infinite product, mainly just a definition, which leads me to ask a question: how do we define convergence of an infinite product in constructive mathematics? The definition seems to depend on decidability of =0=0.

    • have added to conjugation action a detailed exposition of how the conjugation action is the internal hom of actions, here.

    • added to invariant a section, here, with detailed exposition of how invariants are equivalently sections of the action groupoid projection

    • started an entry looping and delooping in an attempt to bring statements together in one place that are currently a bit scattered over the nnLab. Not done yet, but need to quit now.