Loading [MathJax]/jax/element/mml/optable/BasicLatin.js
Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • 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 nLab. 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.

    • 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.

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

    • Someone anonymous set up a file with ’Home Page/solution set condition’ as its name. It was empty so I have renamed it empty 145.

    • created stub for WZW model in reply to a question by a colleague (on how to obtain the Hilbert space of states in that model)

    • I restructured the (stubby) entry higher geometry a bit, following the logic of big and small toposes.

      The point being: you can

      • either axiomatize geometric structure on a little -topos. That’s what the definition of structured (infinity,1)-toposes aims to achieve.

      • or axiomatize geometric structure in a big -topos. That’s what the definition of cohesive (infinity,1)-topos aims to achieve.

    • I keep trying to see to which degree one may nail down supergeometry via the yoga of adjoint modalities.

      My the starting point (maybe there is a better one, but for the time being that’s the best I have come up with) is that the inclusion of commutative algebras into supercommutative algebras is reflective and coreflective, the reflector quotients out the ideal generated by the odd part, the coreflector picks the even subalgebra.

      Passing to sheaves over the sites of formal duals to algebras, this gives an adjoint modality which (and you may not like that now, but keep in mind that it’s just notation which doesn’t reall matter) I decided to denote

      (The mnemonic is this: in Feynman diagrams is the symbol for the bosonic particles, so that denotes taking the bosonic subspace. Similarly a single fermion in a Feynman diagram appears as , so fermion bilinears look like and that has to suffice to remind you of general even numbers of fermions.)

      The details (there are not many, it’s straightforward) are at super formal smooth infinity-groupoid.

      Now, while nice, this falls short of characterizing supercommutative algebras.

      For instance the inclusion of commutative algebras into commutative algebras equipped with /n-grading (for any 2n<, with no extra signs introduced when swapping factors) works just as well. (For the inclusion into -graded algebras there are also left and right adjoints but they coincide, and so I declare that this case is excluded by demanding faithfulness/non-degeneracy of the model.)

      So the question is, which natural-looking further conditions could we impose on the above adjoints such as to narrow in a bit more on supercommutative algebras?

      Here is one observation:

      with the setup as described at super formal smooth infinity-groupoid we also have the reduction modality which, on function rings, takes away the nilpotent ideal in an algebra, and its right adjoint, the de Rham stack functor :

      .

      For both supercommutative algebras and for commutative algebras with grading we have inclusions of images of these functors

      R

      On the other hand, for supercommutative algebras but not for commutative algebras with grading, there is also an inclusion of images diagonally

      . \rightsquigarrow \Im \; \simeq \; \Im \,.

      Because evaluating this via Yoneda on representables, then by adjunction this means that on these

      \Re \rightrightarrows \; \simeq \; \Re

      hence that the reduced part of the even part is the reduced part of the full algebra.

      But this says that the odd part of the algebra is nilpotent! This is something that is true for supercommutative algebras, but which need not be true for any old commutative algebras with grading.

      It is tempting to say that this is a kind of Aufhebung , though it differs from what is currently discussed in that entry in that the diagonal inclusion is along nw-se instead of ne-sw. But I think that just means the entry should be generalized.

      In summary, while the Aufhebung-condition here gets closer on narrowing in the axioms on supercommutative algebras, it still does not quite characterize them. So far these axioms still allow in particular also (sheaves on formal duals to) cyclically-graded algebras whose non-0 graded parts are nilpotent.

      The first question seems to be: do we have further natural-looking axioms on the modalities that would narrow in further on genuine supergeometry?

      But possibly a second question to consider is: might there be room to declare that the further generality allowed by the axioms is something to consider instead of to discard. If there is a nice axiomatics that characterizes something a tad more general than supergeometry, maybe that’s indication that this generalization is of interest?

      Not sure yet.

    • I have added the pair of references

      • John Francis, Derived algebraic geometry over n\mathcal{E}_n-Rings (pdf)

      • John Francis, The tangent complex and Hochschild cohomology of n\mathcal{E}_n-rings (pdf)

      to the References-section at various related entries, such as at derived noncommutative geometry.

      (Thanks to Adeel in the MO-comments here. He watched me ask the question there on three different forums, before then giving a reply on the fourth ;-)

    • Someone created a page Gabriel-Zisman. I have changed the name to Gabriel and Zisman and made a start on a (stub) entry.

    • Just added a references to the entry Morita equivalence. Noticed that the entry is in a woeful state.

      Can’t edit right now, but hereby I move some ancient query boxes that were sitting there from there to here:


      — forwarded query boxes:

      +–{: .query} David Roberts: More precisely, a Morita morphism is a span of Lie groupoids such that the 'source leg' has an anafunctor pseudoinverse. Anafunctors are only examples of Morita morphisms, in the sense that open covers U iM\coprod U_i \to M are examples of surjective submersions.

      I’m also not sure that this should be called the folk model structure, as I don’t think it exists for groupoids internal to DiffDiff. Details of the model structure are in a paper by Everaert, Kieboom and van der Linden, but seem to be tailored towards groupoids internal to categories of algebraic things (e.g semiabelian categories). I think the best one can do is a category of fibrant objects, but that is not something I’ve looked at much.

      Toby: For me, an anafunctor involves a surjective submersion rather than an open cover, which is how that got in there. The important thing is to have equivalent hom-categories.

      David Roberts: I’m not what I was thinking at the time, but you are pretty much right: the definition of anafunctors depends on a choice of a subcanonical singleton Grothendieck pretopology, so it was remiss of me to demand the use of open covers :) As for the definition of Morita morphism, I now can’t remember if that referred to the span which is an arrow in the localised 2-category or the arrow in the unlocalised 2-category that is sent to an equivalence under localisation. At least for me, the terminology Morita morphism evokes the generalisation from a Morita equivalence (a span of weak equivalences) to a more general morphism in that setting.

      Toby: I think that a Morita morphism should be a span, although now I'm not sure that this is what the text says, is it? I should check a reference and then change it.

      David Roberts: I’m fairly sure I’ve heard Lie groupoid people (Ping Xu springs immediately to mind) speak about a Morita morphism as being a fully faithful, essentially surjective (in the appropriate sense) internal functor, but I disagree with their usage. If this is indeed the case, we could note the terminological discrepancies =–

      +– {: .query}

      So is it true that there is a model category structure on algebras such that Morita equivalences of algebras are spans of acyclic fibrations with respect to that structure?

      Zoran Škoda: Associative (nonunital) algebras make a semi-abelian category, ins’t it ? So one could then apply the general results of van den Linden published in TAC to get such a result, using regular epimorphism pretopology, it seems to me. It is probably known to the experts in this or another form.

      =–

    • I fixed two grey links by creating pages for Whitehead’s two papers on Combinatorial Homotopy. If I get around to it I may add in a table of section headings with some comments. The entries are stubs at the moment.

    • I gave “adjoint cylinder” its own entry.

      This is the term that Lawvere in Cohesive Toposes and Cantor’s “lauter Einsen” (see the entry for links) proposes for adjoint triples that induce idempotet (co-)monads, and which he proposes to be a formalization of Hegel’s “unity of opposites”.

      In the entry I expand slightly on this. I hope the terminology does not come across as overblown. If it does, please give it a thought. I believe it is fun to see how this indeed formalizes quite well several of the examples from the informal literature.

      I am not sure if “adjoint cylinder” is such a great term. I like “adjoint modality” better. Made that a redirect.

    • I added two links to interesting looking ArXiv papers at Snigdhayan Mahanta. I also updated the link to his webpage which was no longer valid.

    • I have been adding stuff related to Green-Schwarz sigma models on super-AdS target spaces – added a list of references here – , and in the course of this touched a bunch of related entries and created a few more minimal entries.

      These references show (more or less) that the relevant super Lie algebra cocycles all lift through the Inönü-Wigner contraction from the super anti-de-Sitter/superconformal group to the super Poincaré-group. What I am really after is seeing which of the cocycles lift to super Möbius space, i.e. not to the coset of the super adS/superconrormal group by a super-Lorentz subgroup, but by a parabolic conformal subgroup. This I don’t understand yet at all, but it might as well be trivial, and I wouldn’t see it at this stage.

      Anyway, I have touched/created

      local and global geometry - table, anti de Sitter group, superconformal group, super anti de Sitter spacetime

      and maybe others, too. At conformal group I have added alightning statement of the key isomorphism with a pointer to the literature.

    • I am still after getting a more fine-grained idea of how to best and systematically attach words (notions) to the pluratily of structures encoded by adjoint pairs of modal operators.

      Over in the GoogleGroup “nLab talk” Mike and I were discussing this in the thread called “modal and co-modal”, I am now moving this to here.

      This issue is to a large extent just a linguistic one, and so should maybe better be ignored by readers with no tolerance for that.

      Myself, I may be very slow in discussing this, as I come back to the question every now and then when something strikes me. For the moment I am just looking more closely at our examples to explore further what in their known situation is natural terminology.

      Here is one random thought:

      given a sharp modality, then we look for the “concrete” objects XX for which the unit XXX \hookrightarrow \sharp X is a monomorphism. In a sense the anti-modal objects for which X*\sharp X\simeq \ast are at the “opposite extreme” of these where a concrete object is all “supported on points”, a sharp-anti-modal object is maximally not supported on points.

      By the way, regarding just the terminology for this special case: I have come to think that this is what should be referred to by “intensive quantity” and “extensive quantity”. In modern language an extensive quantity in thermodynamics is a differential form in positive degree (e.g. the mass density 3-form on Euclidean 3-space) while an intensive quantity is a 0-form, hence a function (for instance a temperature function on Euclidean 3-space).

      Now the sheaf of functions \mathbb{R} is indeed such that \mathbb{R}\hookrightarrow \sharp \mathbb{R}, while the sheaves Ω p\mathbf{\Omega}^p of differential forms in positive degree are indeed such that Ω p*\sharp \mathbf{\Omega}^p \simeq \ast. The objects XX with X*\sharp X\simeq \ast are “extensive” in the literal sense that they “extend further than a single point”. This is much the “extension” by which also Grassmann’s Ausdehnungslehre refers to his forms.

      A related random thought: so given any modal operator \bigcirc we should probably be looking at objects XX such that XXX \to \bigcirc X is a mono, or is nn-truncated for general nn.

      For instance the “separated objects” for the constant *\ast-modality are the mere propositions. Generally, the nn-truncated objects are maybe more naturally thought of as being part of what the *\ast-modality encodes. From this perspective the nn-truncation monads on the one hand and the monads appearing in cohesion on the other play somewhat different roles in the whole system, which maybe explains some of the terminological mismatch (if that’s what it was) that Mike and I were struggling with in the earlier thread.

    • I tried to slightly polish the article Dold--Kan correspondence (section: Details) to reduce my confusion concerning the concept of "normalized Moore complex" for a simplicial abelian group. Before I did this work, two different definitions of this concept were given, and it wasn't made vividly clear that these two definitions are naturally isomorphic. In fact that's made clear on the page Moore complex, but I didn't think to look there at first. So, I've tried to make this page a bit more clear and self-contained.

      This page could still use lots of work. There's a half-completed proof, and I suspect the notation and terminology regarding "normalized Moore complex", "alternating face complex", etc. is not completely consistent throughout the $n$Lab.

      I'm also quite sure that in the abelian case, about 100 times as many people have heard of the alternating face complex --- except they call it the "normalized chain complex" of a simplicial abelian group. It may be okay to force people to learn about the normalized Moore complex, which has the advantage of applying to nonabelian simplicial groups. But, it's good to explain what's going on here.