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stub for WZW-type superstring field theory
stub for Seiberg duality motivated from this discussion at “Theoretical Physics Stack Exchange” (is this now publically visible?)
felt the need for 0-morphism
New entry directional derivative, redirecting also Gâteaux derivative. Much of the material is adapted from Fréchet space (which also calls for this entry). Somebody should write more on the (possibly infinite-dimensional) manifold case.
Somebody signing as “Stephan” has made half a dozen or so edits lately. Does anyone know who this might be? Because I would like to suggest to him to announce his changes here.
Mostly they were very useful corrections. But at nice category of spaces and at groupoid object in an (infinity,1)-category I felt that the comment added there by him was in need of a bit of rephrasing. Nothing serious, but I’d like to know who he is to sort this out.
He also, I think, created a new entry titled Principal bundles, groupoids and connections
I have cross-linked the two entries homotopical algebra and higher algebra.
At homotopical algebra I moved the text that had existed there into a subsection “History”, because that’s what it is about, right? I added a section “Idea” but so far only included a link to higher algebra there. We could maybe merge the two entries.
added to connection on a bundle
a Definition (nPOV-flavor, of course)
a Properties-section with statement and proof of the fact that every bundle does admit a connection.
started something called table of orthogonal groups and related and included it into the relevant entries
stub for dg-manifold and dg-scheme
I have started creating a table of branes and their worldvolume theories . So far it looks as follows
I have created all the missing entries to complete this and have included the table in all relevant entries.
I am starting an entry 7-dimensional supergravity in order to collect some references that I need
I have been adding to AdS/CFT in the section AdS7 / CFT6 a (of course incomplete) list of available evidence for what is going on.
This is triggered by the fact that we have a proposal for a precise formalization of the effective 7d theory.
stub for configuration space with -topos theoretic definition. See also phase space
I have added an explanatory paragraph to n-poset in reply to this MO question.
Also, at poset itself I have added a word (“hence”) to indicate that if something is a category with at most one morphism between any ordered pair of objects, then it is already implied that if there are two morphisms back and forth between two objects, then these are equal.
I have split off an entry epi/mono factorization system in order to better be able to amplify the higher pattern that this sits in
added to homotopy image a brief remark in a new subsection on how this is given by the n-connected/n-truncated factorization system for .
at n-connected/n-truncated factorization system I have created an Examples-section with a brief indication of what this factorization “means” for low values of (from to ).
I plan to redo measurable space, and the outline of the plan is now at the bottom.
For the nonce, I’ve moved some material to a new article sigma-algebra, and some of that thence to the previous stub Borel subset.
Stubby beginnings of articles on well-quasi-order, antichain, and graph minor. Some minor mention (ha ha) of the Robertson-Seymour theorem. Please feel free to add more.
new section at symplectic infinity-groupoid on Hamiltonian vector fields on symplectic oo-groupoids.
I have significantly extended the list of references at geometric Langlands program. Langlands’ is here in the role of adjective and geometric Langlangds is an informal abbreviation. I have changed the former name geometric Langlands to geometric Langlands program but due cache bug now two pages seem to exist in parallel.
On Sep 29, I added the new stubs collective field theory and large N limit. My interest is in the question I just posed at theoreticalPhysics.stackexchange systematic-approach-to-deriving-equations-of-collective-field-theory-to-any-order.
In my opinion/wish, I should have been better prepared to ask that question (in depth reading of some of the key references are on my todo list), but I posted the question a bit earlier than ready for a better documented question, as anyway there is a need of constructive kick-off of the activity at thPh.stExch.
I wrote about the boolean algebra of idempotents in a commutative ring. There’s also stuff in there about projection operators (that page doesn’t exist).
Updated the reference to "The Hunting of the Hopf Ring" since it's now appeared in print.
I added a comment to the end of the discussion at predicative mathematics to the effect that free small-colimit completions of toposes are examples of locally cartesian closed pretoposes that are generally not toposes.
I added the notion of a regular curve to curve. In differential geometry, for most purposes only regular curves are useful: the parametrized smooth curves with never vanishing velocity. Smooth curves as smooth maps from the interval are not of much use without the regularity condition: their image may be far from smooth, with e.g. cusps and clustered sequences of self/intersections.
added stuff to Lie 2-group: more in the Idea-section, more examples, some constructions, plenty of references.
I rephrased the classical alternative formulations at well-founded relation to define relations with no infinite descent.
I am working on further bringing the entry
infinity-Chern-Weil theory introduction
into shape. Now I have spent a bit of time on the (new) subsection that exposes just the standard notion of principal bundles, but in the kind of language (Lie groupoids, anafunctors, etc) that eventually leads over to the description of smooth principal oo-bundles.
I want to ask beta-testers to check this out, and let me know just how dreadful this still is ! ;-) The section I mean is at
I found an interesting question on MO (here) and merrily set out to answer it. The answer got a bit long, so I thought I’d put it here instead. Since I wrote it in LaTeX with the intention of converting it to a suitable format for MO, it was simplicity itself to convert it instead to something suitable for the nLab.
The style is perhaps not quite right for the nLab, but I can polish that. As I said, the original intention was to post it there so I started writing it with that in mind. I’ll polish it up and add in more links in due course.
The page is at: on the manifold structure of singular loops, though I’m not sure that that’s an appropriate title! At the very least, it ought to have a subtitle: “or the lack of it”.
created model structure on cosimplicial simplicial sets, mainly in order to record two references.
added today’s article by Monnier to the references at quantum anomaly and self-dual higher gauge theory
quick stubs for Killing tensor and Killing-Yano tensor in reply to a question here
at geometric quantization of symplectic infinity-groupoids (which currently still redirects to symplectic infinity-groupoid) I am beginning to add some genuine substance. So far:
an Idea-section
and a section Prequantum circle (n+1)-bundle with the general abstract definition and the beginning of some examples
I created Frobenius map, since I had linked to it in several places.
added to Chern-Simons element in the section Standard Chern-Simons form a detailed pedestrian proof of how the standard Chern-Simons 3-form is reproduced from this machinery.
On well-order and elsewhere, I’ve implied that a well-order (a well-founded, extensional, transitive relation) must be connected (and thus a linear order). But this is not correct; or at least I can’t prove it, and I’ve read a few places claiming that well-orders need not be linear. So I fixed well-order, although the claim may still be on the Lab somewhere else.
Of course, all of this is in the context of constructive mathematics; with excluded middle, the claim is actually true. I also rewrote the discussion of classical alternatives at well-order to show more popular equivalents.
Created F-category and rigged limit.
Created factorization structure for sinks, and added remarks to Grothendieck fibration, M-complete category, and topological concrete category about constructing them. Largely I wanted to record the proof of the theorem that in an factorization structure for sinks , the class necessarily consists of monomorphisms. It’s a nice generalization of Freyd’s theorem about complete small categories, which has more of the feel of a useful theorem than of a no-go theorem like Freyd’s.
At first I thought that the lifting of factorization structures described at topological concrete category would work for solid functors too, but then I couldn’t see how to do it. Does anyone?
[Jim Stasheff means to post the following message here to the Forum, but accidentally (I think) posted it here instead (I guess because that is the forum that comes up when one googles for “nForum”)]:
Solutions of the KZ equations are usually given in terms of assymptotic behavior in certain regions. Since the region on which the eqns are defined has a nice compactication, what is the obstruction to extending the solutions to the compactification?
I started a stub
and added a bit of overlapping material to quandle. I would like to talk about Lie and Jordan triple systems, but I need this introductory material first.
Created M-complete category.
created
(to go with the discussion at ∞-gerbe).
There are other notions of “center” of -groups. But this is one of them.
Inspired by Todd’s work at well-founded relation, I’ve written simulation.
I made an initial foray into explaining the coalgebraic aspects of recursion schemes (following Taylor) by editing well-founded relation, by including a new section “Coalgebraic formulation”. (The title is slightly awkward when it appears just after the section “Alternative formulations”; that section was on alternative formulations which are possible in classical logic, whereas this section is on a different language for presenting the intuitionistic case. Therefore I didn’t want to make it a subsection of “Alternative formulations” as currently written.)
Also some words there on the coalgebraic formulation of simulations.
Edit: I decided to rename “Alternative formulations” by “Formulations in classical logic”; I hope no one minds.
Inspired by Tom Leinster’s recent blog posts, I have created Hausdorff metric, and added the metric-space version (sans the categorical interpretation, for now) to geodesic convexity.
I have started a page compactly generated model category. But this is at best “under construction” (I have added a warning). I first just wanted to record Jardine’s definition referred to there. But I find something weird at least in the notation he has (he must mean homotopy colimits in the simplicial localization instead of plain colimits on homotopy hom-sets?!), and I don’t seem to find two different autrhors that agree on all the ingredients.
I’ll leave the entry in this unsatisfactory state for the moment. There is a warning.
created gradient
created 2-gerbe.
This is brief, but comes with a quick remark on the two possibilities of the definition, one as in Breen, one as in Lurie.
(And not to speak of bundle 2-gerbes, which is really a very different definition alltogether.)
I also edited infinity-gerbe accordingly, pointing out how Lurie’s definition is just a special case of what one would arrive at if one went with Breen and kept increasing .
I’ve moved part of my beginner’s summery on Mac Lane’s proof of the coherence theorem for monoidal categories to the Lab.
Surely there are many mistakes, possibly fatal: I am most worried about the naive definition I used for the syntax of arrows (sadly, I could only use my nearly zero knowledge of logic), and the part including the units and unitors. There are probably many itex errors too.
At some point I’ve realized that it was silly to use the cumbersome arrow language as is (say, writing a string like instead of , etc.), but to correct this required changing all figures, and this is too much for me at the moment. I also apologize for using files for figures (admiting that some of them shouldn’t even be figures), but it was too much work to do all the transition from my notes to itex in one jump.
I hope learning by seeing this page modified by people knowing more than me (anyone here, that is :) ), but if this page seems beyond hope, I have no objections to renaming it so that it will not clutter the ’lab.
By the way, now that I’ve re-entered the page, all figures appear with a question mark, and to the see the figure I have to press on the question-mark link (previously I’ve seen all figures appearing in the page). Is there any way to solve this problem?
To add recent surge of activity (by Urs, me etc.) in Lab on symplectic geometry, variational calculus and mechanics I created the entry Lagrange multiplier following mainly Loomis-Strenberg. For convenience, I uploaded the critical 4 pages from their book.
I am starting an entry symplectic infinity-groupoid.
This is still in the making. Currently there are two things:
A little general indication of what this is supposed to be about;
A proof of an assertion that serves to justify the whole concept.
Namely, the literature already knows the concept of a symplectic groupoid. This plays a big role in Weistein’s program and in particular in geometric quantization of symplectic groupoids, which induces, among other things, a notion of geometric quantization of Poisson manifolds.
As far as I am aware (though I might not have been following the latest developments here, would be grateful for comments) it is generally expected that symplectic groupoids are formally the Lie integration of Poisson Lie algebroids, but there is no proof or even formalization of this in the literature.
In the entry I indicate such a formalization and give the respective proof.
The idea is that this is a special case of the general machine of infinity-Chern-Weil theory:
namely: the symplectic form on a symplectic Lie -algebroid such as the Poisson Lie algebroid is Lie theoretically an invariant polynomial. So the -Chern-Weil homomorphism produces a corresponding morphism from the integrating smooth -groupoid to de Rham coefficients. This is a differential form in the world of smooth -groupoids.
The assertion is: this comes out right. Feed a Poisson Lie algebroid with its canonical invariant polynomial into -Chern-Weil theory, out comes the “classical” symplectic Lie groupoid.
(I do this for the case that the Poisson manifold is in fact itself symplectic, which is the only case I remember having seen discussed in earlier literature. But I think I can generalize this easily.)
started something at higher symplectic geometry.
Not much there yet, more a kind of announcement for the moment. But it may serve the purpose of providing a sentence on in which sense multisymplectic geometry and symplectic Lie n-algebroids are two aspects of a more general thing.
I addede a paragraph at Poincaré duality about the generalizations, and created the entry (so far only descent bibliography) Grothendieck duality; the list of examples expanded at duality. All prompted by seeing the today’s arXiv article of Drinfel’d and Boyarchenko.
stub for Hadwiger’s theorem