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Marc Hoyois created formally real field back in August (which was never announced here), and now I've created formally real algebra (and linked them to one another).
Stub symplectic integrator, just a list of basic references so far, redirecting aslo multisymplectic integrator.
have added a paragraph tangent infinity-category – Tangent infinity topos meant to extract the argument from Joyal’s “Notes on Logoi” that the tangent -category of an -topos is an -topos. Then a remark on how this should imply that the tangent -topos of a cohesive topos is itself cohesive over the tangent base -topos.
I am not making any claims tonight, just sketching an argument. Hope to come back to it tomorrow when I am awake again.
I have started a stub for smooth super infinity-groupoids, with the evident definition and observation that this is cohesive, but nothing else so far. To be worked on. (similar to locally-contractible infinity-groupoid)
I’ve constructed the page p-divisible group since I need it for my height of a variety page. I have to admit that I’m incredibly embarrassed that no matter how many times I look up the words “directed” “inductive” “projective” “limit” “colimit” etc I never seem to use them correctly. All of the systems are as I showed I thought this corresponded to directed, inductive, or colimit, but when I looked up inductive limit in the nlab it seemed to be indicating the opposite, so maybe some of the uses are wrong.
In the References-section at 2-sheaf I have added three “classical” references:
in the 1970s Grothendieck, Giraud and then Bunge usually considered “2-sheaves” – namely category-valued stacks – by default. Also there is a good body of work on 2-sheaves realized as internal categories in the underlying 1-sheaf topos. I have added a pointer to Joyal-Tierney’s Strong stacks so far, but I think much more literature exists in this direction.
But if one goes this internalization-route at all, what one should really do is, I think, consider weak internal categories in the (2,1)-topos over the underlying site.
Has this been studied at all? Does anyone know how 2-categories of weak internal categories in -toposes relate to 2-toposes? At least under nice conditions these should be equivalent, I guess. But I want to understand this better.
brief entry differentiable (infinity,1)-category
That is, Jordan–Banach algebras (although there is actually a distinction between these).
Lagrange inversion, redirecting also Lagrange inversion formula and Lagrange inversion theorem, previously wanted at Lambert W-function, noncommutative symmetric function and at Faà di Bruno formula.
I don't know why we never had endofunction, but we didn't; now we do.
I have made functional and operator primarily about the meanings of these in higher-order logic, where these terms are used exclusively and unqualified. I have accordingly split off linear functional from functional; linear operator (redirecting to linear map) was already separate from operator (which was only for disambiguation). I have also checked each incoming link to functional or operator (or a plural form) to link instead to linear functional or linear operator when appropriate.
That said, there are such things as nonlinear functionals and operators on abstract vector spaces, things which are also not functionals or operators in the type-theoretic sense. Possibly we would want pages such as nonlinear functional and nonlinear operator to cover these. (Compare nonassociative algebra, which covers a topic more general than what is covered at associative algebra but also could not be covered at simply algebra.)
I did not know what to do with the phrase ‘various discretised versions are interesting in finite geometries as well as numerical analysis’. Are these linear functionals, type-theoretic functionals, both, or neither?
The revolution will not be televised, but it will be wikified at power-associative algebra.
I created polarization identity and added some disambiguation to polarization.
With our “String Geometry Network” we have another meeting in October at the Max-Planck Institute for Mathematics in Bonn.
In each such meeting we have, besides research talks and discussion sessions, a kind of “reading course”, something to get us all on the same page of some topic.
This time the idea is to talk about higher supergeometry and “super-string geometry”, if you wish. I am preparing some notes to go with this, and naturally I got inclined to prepare them on the nLab. They will be developing here in the entry
Currently there is just an introduction and then a session outline with just a few linked keywords. I’ll be developing this as days go by. Depending on which reactions I get, there might be drastic revisions, or just incremental extension. We’ll see.
started Yetter model, still a stub so far. Tim, I trust you will add references?! :-)
New entry: Reedy category with fibrant constants.
Circumstances prompted me to write a kind of pamphlete pointing out some aspects that seem worth taking notice of have not found much appreciation yet:
This surveys how basic theorems about the standard foundation of quantum mechanics imply an accurate geometric incarnation of the “phase space in quantum mechanics” by an order-theoretic structure that combines with an algebraic structure to a ringed topos, the “Bohr topos”. While the notion of Bohr topos has been motivated by the Kochen-Specker theorem, the point here is to highlight that taking into account further theorems about the standard foundations of quantum mechanics, the notion effectively follows automatically and provides an accurate and useful description of the geometry of “quantum phase space” also in quantum field theory.
created Clebsch-Gordan coefficients
I added Sinh’s thesis plus a link to a scan to 2-group
brief entry “daseinisation”
(Note: I am not embracing the term, I just happen to want to record that somebody proposed it.)
brief entry for spectral presheaf
Zoran started enveloping von Neumann algebra earlier this month; I've added more characterizations and something about uses.
There was already a discussion of ends in the topological sense at proper homotopy. (I had never seen the term hemicompact before. I knew of -compact which is almost the same.)
I’d like to add some sentences describing a geometric understanding of the “torsion” of a connection on a Riemannian manifold to torsion. But perhaps my understanding of torsion is wrong, so I’m running it by you guys first. I wrote it down on math overflow and I’m curious what people think.
Added a link to an expository talk I gave on “The geometry of force” giving an elementary explanation of the classical Kaluza-Klein mechanism (i.e. the idea that geodesics on the principal bundle project down to curved trajectories on base space apparently experiencing a “force”). Following the book of Bleecker, Gauge theory and variational principles.
This came up in another thread, where it was not really on-topic. I want to re-post it here so that it gets due attention and maybe finds a resolution:
I find the content at canonical morphism unsatisfactory to the extent that I am voting to replace it by something else.
First why I find it unsatisfactory: the entry describes a proposal by Jim Dolan from some time ago for how to give a formal meaning to the colloquial use of “canonical” in mathematics, but it seems to me that
nobody, not even Jim Dolan himself, ever used that in practice;
it does not actually capture much at all about the colloquial use of “canonical” (see below for a proposal of mine of what a proper formalization would involve);
the only almost-application mentioned at the bottom of the entry, which is about morphisms of QFTs, has nothing of the “canonical” flavor to it at all (on the contrary!), and the curious notion discussed there actually becomes natural if one instead considers QFTs with boundaries, as formalized on the last pages of Lurie’s “classification of TFT” article.
In summary, I find that the article gives an ill-motivated definition which is actually misleading and has no support from usage/practice.
If this sounds harsh, please take it as motivation to convince me otherwise. I’ll be happy to be convinced by a good argument and will use it to improve the entry accordingly.
But currently I think the entry content should be replaced with something else. I would tend to think that a formalization of “canonical” should involve something as follows, instead.
It should involve some notion of constructiveness. What is canonical, in colloquial meaning of the word, is that which we can actually construct, with given data (given terms).
For instance for just any set without further information, the reason why is the canonical map from to itself is because this is the only one we can actually name, whose term we can actually construct. There are all these other maps, but we can’t actually name them with the given information.
Or: the reason why are the canonical coordinates on the plane is because they are the two which one can actually construct given the data by which the plane was constructed, namely the two projection maps out of the project. The reason why all the other coordinates that we might put on are “not canonical” is that while they “exist” in the sense of existence of mathematics, we cannot actually construct them with the given data.
I expect that somebody with more genuine type-theoretical practice can easily see what I am getting at here and maybe give it a more pronounced formulation.
added to global model structure on functors that theorem that the projective and the injective global model structure on functors with values in a combinatorial model category is itself again a combinatorial model category.
Started Dwyer map.
I added a section Hom-spaces between cofibrant/fibrant objects with a few lemmas and their proofs at
(infinity,1)-categorical hom-space.
(The proofs are intentionally very small-step and hopefully "pedagogical".)
I also reworded the introduction part a bit and replied further in the old query box there.
Effectively my point is: I am not overly happy with the title of that entry myself, but the alternatives proposed so far still strike me as worse. The main deficiency of the title is that it may sound a bit awkward. But it has the advantage of being fairly accuratively descriptive.
But I won't be dogmatic about this. If there is a wide-spread desire to rename the entry, please feel free to do so.
created an entry Hamiltonian n-vecotr field (redirecting Hamiltonian multivector field) with the definition of the “de Donder-Weyl”-Hamiltonian flows of -volumes and, secondly. with some comments on how to interpret this in higher geometry. Will further expand on this second piece a little later, need to interrup now.
I have split off reduced phase space from covariant phase space and started to expand a bit.
In particular I tried to highlight a bit the important point that the exact presymplectic form which is induced by any local action functional on its covariant phase space (as discussed there), still has to be equipped with equivariant structure as a U(1)-prinicipal connection in order to pass to the reduced phase space.
This is an obvious point that however I find is glossed over in much of the literature and leads to some confusion in some places: some literature fond of the covariant phase space-construction from local action functionals will highlight that this always has exact presymplectic form and will take this as reason to disregard all the subtleties of geometric quantization, which pretty much disappear for exact (pre-)symplectic forms. The point missed in such discussion is that there is non-trivial equivarint structure on the prequantization of this presymplectic form.
This subtlety as such is of course treated correctly in all of the mathematical literature listed at qauzntization commutes with reduction, of course. But that literature in turn doesn’t mention the important construction of covariant phase spaces from local Lagrangians.
Therefore, if anyone can point me to references that do BOTH of the following:
discuss the covariant presymplectic phase space induced form a local Lagrangian;
discuss the need to put equivariant connection structure on the canonically induced globally defined presymplectic potential;
I’d be grateful.
To complement the entry on Chu construction, I have created a new entry on Chu spaces, simple examples, which I hope to link up with the formal concept analysis entry.
I created localizable measure to record the definition. Needs more properties.
I created opposite poset to satisfy some links.
Someone has created www.maths.ed.ac.uk/~aar/papers/witten.pdf. This seems to me a strange thing to do. It is perhaps someone just trying out code.
am making some notes at off-shell Poisson bracket, based on a discussion that I am having with Igor Khavkine
gave extended supersymmetry its own entry (the content had previously been scattered around in other entries)
brief entry for Velo-Zwanziger problem
added to Donaldson theory a pointer back to Lagrangian correspondences and category-valued TFT
at Peierls bracket I have added a bunch of further references and have slightly expanded the Idea-section
I have split off an entry geometric quantization of symplectic groupoids from symplectic groupoid.
I have also tried to clean up and make more systematic the Idea-section at geometric quantization
I have now what should be a readable pre-version of
This still needs a round of polishing or two. But it should be at least readable.
If anyone is interested, have a look, be critical and try to poke holes into it.
added some references to lattice field theory and lattice renormalization. Though I am still looking for a reference on the latter that states the basic mechanism clearly…
at superselection sector I tried to state more explicitly the fact that superselection sectors are equivalently the irreps of the algebra of observables in the space of quantum states.
Also cross-linked with irreducible representation and with quantum observable.
over at prequantized Lagrangian correspondence I needed to point to Newton’s laws, so I have started an entry Newton’s laws of motion. So far mainly it just contains briefly the content of Newton’s laws in modern language.
prompted by this physics.SE question I ended up adding to Idea-section of the entry quantization a new subsection titled Motivation from classical mechanics and Lie theory .
Changed the page local section to discuss a slightly more general concept than the local sections of a bundle, and over a more general pretopology.
added some references on representationology of to F4
added to geometric representation theory a quote of program description of the MSRI program next year.
had need for a small table worldvolume-target supersymmetry of brane sigma-models and so I created one. Have included it into the relevant entries.
Also created a stub for superembedding approach, the middle entry of the table.
(An nForum issue: as of late I get to see the nForum only in its plain HTML-form, which is very inconvenient. Is this a problem just on my side, or does anyone else experience this?)
As we were discussing profinite completions the other day in another thread I thought I would add in some points about completed group algebras at profinite group and add some mention of pseudo compact algebras to the pre-existing entry on pseudocompact rings.
It is not clear to me what the connection between these algebras and profinite algebras should be. These pseudocompact and related linear compact algebras use finite dimensionality instead of finiteness to get a sort of algebraic compactness condition.
created super Poincare Lie algebra
linked to it from super Euclidean group and from supergravity
I have Polchinski’s textbook a category:reference-entry String theory, for purposes of better linking to it
started a topic cluster table of contents higher spin geometry - contents and included it as a “floating table of contents” into relevant entries
Started locus following this discussion.
Should left exact localization link to reflective sub-(infinity,1)-category?
I added a link to the published version of my notes on universal simplicial bundles, here on Urs’ web.
brief remark charge conjugation matrix, just because I needed to be able to point to it
added little bit more to super translation Lie algebra, including a remark that it is a central extension of the superpoint, regarded as an abelian super Lie algebra.