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After contributing to the article on parallelogram identity, I added to isometry and created Mazur-Ulam theorem. The easy proof added at isometry, that shows an isometry between normed vector spaces is affine if is strictly convex, might lead one to suspect that the proof under parallelogram identity was overkill, but I think that’s an illusion. Ultimately, I believe the parallelogram identity is secretly an expression of the perfect ’roundness’ of spheres, connected with the fact observed by Tom Leinster recently at the Café that the group of isometries for the norm is a continuum, whereas for other in the range , you get just a finite reflection group (this is for the finite-dimensional case, but there’s an analogue in the infinite-dimensional case as well).
The Mazur-Ulam theorem removes the strict convexity hypothesis, but adds the hypothesis that the isometry is surjective. The conclusion is generally false if this hypothesis is omitted.
I have started a separate entry on strong shape theory, but it is only a stub with references filtched from shape theory.
I have split off universal quantifier and existential quantifier from quantifier in order to expose the idea in a more pronounced way in dedicated entries.
Mainly I wanted to further amplify the idea of how these notions are modeled by adjunctions, and how, when formulated suitably, the whole concept immediately and seamlessly generalizes to (infinity,1)-logic.
But I am not a logic expert. Please check if I got all the terminology right, etc. Also, there is clearly much more room for expanding the discussion.
Thought I’d nick an another answer from MathOverflow and paste it to the nLab. Unfortunately, doing an internet search for “functional analysis type” or even cotype doesn’t look like I’m going to be able to figure out what those terms mean all that quickly …
Oops. Forgot the link: isomorphism classes of Banach spaces.
Added parallelogram identity since it was linked from type (functional analysis). Actually, I mostly stole the content from another wiki, (this one) but I don’t think that the original author will mind.
stub for cohomology of operads, so far just in order to record Charles Rezk’s thesis.
Bill Johnson kindly sent me an explanation of type and cotype for Banach spaces which I’ve mangled and put up at type (functional analysis).
I have created some genuine content at implicit function theorem. I’d like to hear the comments on the global variant, which is there, taken from Miščenko’s book on vector bundles in Russian (the other similar book of his in English, cited at vector bundle, is in fact quite different).
finally a stub for (infinity,1)-semitopos
I have created an entry notions of type to be included under “Related notions” in the relevant entries.
(I have managed to refrain from titling it “types of types”.)
Which notions of types are still missing in the table?
I have added some remarks to chain complex, model structure on chain complexes, homotopy limit, and derivator regarding the fact that every chain complex over a field is equivalent to its homology (regarded as a chain complex with zero differentials).
In reaction to the public demand exhibited by Guillaume Brunerie's comments I have created an entry
To replace some anonymous scribblings, I cribbed some definitions from Wikipedia to get a stub at deformation retraction.
quick note on 2-framing
I thought up until just a few minutes ago that I had proved that WISC was equivalent to local essential smallness of . Mike urged me to put my proof on the lab, but in doing so I discovered it was flawed. So now WISC just has a proof that the principle implies local essential smallness.
I noticed there is no entry electrodynamics so I “created” it as a redirect to the existing stub electromagnetism. Though I personally consider electromagnetism as a phenomenon in real world, while electrodynamics just as a theoretical formalism to describe it, i.e. a theory of electromagnetism. There is some overlap between existing entries, like there is another, rather stubby classical electrodynamics, and entry quantum electrodynamics. The real content is in electromagnetic field.
I am starting something at higher dimensional WZW theory
stub for false vacuum
created cohomology of local net of observables. I have included a brief Idea-section but mainly this is, for the moment, to record references.
New entry combinatorial Hopf algebra. Reference additions or updates in Hopf algebra, BV formalism, Hall algebra, graph homology, Marcelo Aguiar, renormalization.
Have a look at horizon.
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.