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created microlinear space
One thing I might be mixed up above:
in the literature I have seen it seems to say that
$ X^D x_X X^D \simeq X^{D(2)}$
with
$ D(2) = { (x_1,x_2) \in R \times R | x_i x_j = 0} $.
But shouldn't it be
$ D(2)' = { (x_1,x_2) \in R \times R | x_i^2 = 0} $.
?
this is a bare list of references discussing the spin-statistics theorem for non-relativistic particles via their configuration space of points,
to be !include-ed into the References-sections of these entries, for ease of synchronization
added missing pointer to
and pointer to:
starting a stand-alone Section-entry (to be !includeed as a section into D=11 supergravity and into D’Auria-Fré formulation of supergravity)
So far it contains lead-in and statement of the result, in mild but suggestive paraphrase of CDF91, §III.8.5.
I am going to spell out at least parts of the proof, with some attention to the prefactors.
I have added to BPS state and to wall crossing pointers to two introductory lecture notes
This is maybe mainly for entertainment. But don’t forget that for newcomers there is a real issue here which may well be worth explaining:
In mathematics it happens at times that one and the same concept is given two different names to indicate a specific perspective, a certain attitude as to what to do whith such objects.
Here are examples:
A quiver is just a directed graph (pseudograph, to be explicit). But one says quiver instead of directed graph when one is interested in studying quiver representations: functors from the free category on that graph to the category of finite-dimensional vector spaces.
A presheaf is just a contravariant functor. But one says presheaf instead of contravariant functor when one is interested in studying its sheafification, or even if one is just intersted in regarding the category of functors with its structure of a topos: the presheaf topos.
(…)
Added reference
John Arte
the standard bar complex of a bimodule in homological algebra is a special case of the bar construction of an algebra over a monad. I have added that as an example to bar construction.
I also added the crucial remark (taken from Ginzburg’s lecture notes) that this is where the term “bar” originates from in the first place: the original authors used to write the elements in the bar complex using a notaiton with lots of vertical bars (!).
(That’s a bad undescriptive choice of terminoiogy. But still not as bad as calling something a “triple”. So we have no reason to complain. ;-)
I am adding to the Lab further pointers here and there on Witten’s story . Added now to self-dual string a pointer to (Witten 95), which I suppose is the first observation of the self-dual string and its ADE classification.
added the statement (now this prop) that smooth manifolds with boundary are fully faithful in diffeological spaces, with pointer to Igresias-Zemmour 13, section 4.16.
Will add the same to diffeological space.
Inspired by the discussion at directed n-graph and finite category, added some examples and further explanation to computad.
Since the page geometry of physics – categories and toposes did not save anymore, due to rendering timeouts caused by its size, I have to decompose it, hereby, into sub-pages that are saved and then re-!included separately.
With our new announcement system this means, for better or worse, that I will now have to “announce” these subsections separately. Please bear with me.
did some substantial edits on this entry:
gave it more of an Idea-section,
tried to streamline the statement of the lemma
spelled out the proof,
added a discussion explaining how this is about irreps forming a (de-)categorified orthogonal/orthonormal linear basis of the representation ring.
gave representation theory a little Idea-section, then added some words on its incarnation as homotopy type theory in context/in the slice over and added the following homotopy type representation theory – table, which I am also including in other relevant entries:
a bare minimum, for the moment just so as to record this references:
Added a note on triangulability and smoothing to 3-manifold.
Added material about the Hantzsche-Wendt manifold on his website:
Also linked names and added arXiv code.
Created page due to recent creation of Hantzsche-Wendt manifold.
Created page due to recent creation of Hantzsche-Wendt manifold.
Was reminded to create this by the recent creation of Ralf Aurich and Sven Lustig. (The german Wikipedia page is now also available.)
at Beck-Chevalley condition I have added an Examples-section Pullbacks of opfibrations with statement and proof that the diagram of presheaves induced by a pullback of a small obfibration satisfies Beck-Chevalley.
Created bare minimum for orthocompact, which will be expanded later.
Edit: I’m not sure if there is a convention for the title? With or without “topological”? One one hand there are compact space, metacompact space or hemicompact space. On the other hand, there are locally compact topological space or paracompact topological space.
Edit: I’ve now linked orthocompact space on hemicompact space and paracompact topological space as there are lemmata connecting them with each other.
Added theorems I searched for two years ago for MSE/4492531 and MSE/4566624 as well as content from the two other answers and the argument by Eric Wofsey on MSE/4209303. (I’ve not linked the former ones due to me having answered there.)
I also added a remark on locally compact as it has two different definitions in literature (compact neighborhood or compact neighborhood base for every point). Just to be sure there is no misunderstanding.
am finally splitting this off as a stand-alone page (material used to be at cosmological constant and at F-theory). Added a paragraph linking with inhomogeneous cosmology.
There is not so much at the nLab about computational complexity, but I stumbled on a reference to the hardness of computing the order polynomial so I added it to this article (while also taking the opportunity to try yet another notation for finite ordinals!).