Page created, but author did not leave any comments.

]]>brief category:people-entry for hyperlinking references at *AdS-CFT duality* and elsewhere

earlier today I had created a stub for anti de Sitter spacetime

]]>added pointer, here and in related entries, to

- Gary Gibbons, Malcolm J. Perry,
*Black Holes and Thermal Green Functions*, Vol. 358, No. 1695 (1978) (jstor:79482)

being one of the early references considering the computation of black hole radiation in terms of thermal field theory on Wick-rotated spacetimes with compact/periodic “Euclidean time”.

]]>A skeleton

]]>added references by Pronk-Scull and by Schwede, and wrote an Idea-section that tries to highlight the expected relation to global equivariant homotopy theory. Right now it reads like so:

On general grounds, since orbifolds $\mathcal{G}$ are special cases of stacks, there is an evident definition of cohomology of orbifolds, given by forming (stable) homotopy groups of derived hom-spaces

$H^\bullet(\mathcal{G}, E) \;\coloneqq\; \pi_\bullet \mathbf{H}( \mathcal{G}, E )$into any desired coefficient ∞-stack (or sheaf of spectra) $E$.

More specifically, often one is interested in viewing orbifold cohomology as a variant of Bredon equivariant cohomology, based on the idea that the cohomology of a global homotopy quotient orbifold

$\mathcal{G} \;\simeq\; X \sslash G \phantom{AAAA} (1)$for a given $G$-action on some manifold $X$, should coincide with the $G$-equivariant cohomology of $X$. However, such an identification (1) is not unique: For $G \subset K$ any closed subgroup, we have

$X \sslash G \;\simeq\; \big( X \times_G K\big) \sslash K \,.$This means that if one is to regard orbifold cohomology as a variant of equivariant cohomology, then one needs to work “globally” in terms of *global equivariant homotopy theory*, where one considers equivariance with respect to “all compact Lie groups at once”, in a suitable sense.

Concretely, in global equivariant homotopy theory the plain orbit category $Orb_G$ of $G$-equivariant Bredon cohomology is replaced by the global orbit category $Orb_{glb}$ whose objects are the delooping stacks $\mathbf{B}G \coloneqq \ast\sslash G$, and then any orbifold $\mathcal{G}$ becomes an (∞,1)-presheaf $y \mathcal{G}$ over $Orb_{glb}$ by the evident “external Yoneda embedding”

$y \mathcal{G} \;\coloneqq\; \mathbf{H}( \mathbf{B}G, \mathcal{G} ) \,.$More generally, this makes sense for $\mathcal{G}$ any orbispace. In fact, as a construction of an (∞,1)-presheaf on $Orb_{glb}$ it makes sense for $\mathcal{G}$ any ∞-stack, but supposedly precisely if $\mathcal{G}$ is an orbispace among all ∞-stacks does the cohomology of $y \mathcal{G}$ in the sense of global equivariant homotopy theory coincide the cohomology of $\mathcal{G}$ in the intended sense of ∞-stacks, in particular reproducing the intended sense of orbifold cohomology.

At least for topological orbifolds this is indicated in (Schwede 17, Introduction, Schwede 18, p. ix-x, see also Pronk-Scull 07)

]]>I am trying to collect citable/authorative references that amplify the analog of the mass gap problem in particle phenomenology, where it tramslates into the open problem of computing hadron masses and spins from first principles (due to the open problem of showing existence of hadrons in the first place!).

This is all well and widely known, but there is no culture as in mathematics of succinctly highlighting open problems such that one could refer to them easily.

I have now created a section *References – Phenomenology* to eventually collect references that come at least close to making this nicely explicit. (Also checked with the PF community here)

Some weird bugs related to rendering redirects (but not only that):

On logarithm, if you scroll down, the redirects at the bottom aren't parsed but just show up as

`[[!redirects logarithm]]`

and the like. Several links above that don't work either. As a result (I presume) redirects to that page don't work; see inferior limit for a broken link to natural logarithm (or just click on the link in this post). Before I edited logarithm just now (two trivial edits that cancelled exactly), the redirects were correctly invisible, but the links in the bullet list just above were also completely invisible! But if you look at the diff page or the specific revision, then it looks fine.On redirect (which I looked at to make sure that I hadn't mixed the syntax up with MediaWiki again), the displayed examples in the grey boxes show the rendered HTML code as text rather than showing text with markup. Again, the diff page and the specific revision are fine.

These are on up-to-date Firefox Portable on Windows 10, although I was on Windows 7 earlier (and actually it's the same Firefox Portable, storing its data on a server 20 miles away from me but being run by my local machines), originally with ScriptSafe set to allow `ncatlab.org`

and then (with no change in the results) with ScriptSafe disabled.

purge

]]>Fix links.

]]>Page created, but author did not leave any comments.

]]>stub entry for the moment, to satisfy links

]]>am starting something

]]>I am moving the following old query box exchange from orbifold to here.

old query box discussion:

I am confused by this page. It starts out by boldly declaring that “An orbifold is a differentiable stack which may be presented by a proper étale Lie groupoid” but then it goes on to talk about the “traditional” definition. The traditional definition definitely **does not** view orbifolds as stacks. Neither does Moerdijk’s paper referenced below — there orbifolds form a 1-category.

Personally I am not completely convinced that orbifolds are differentiable stacks. Would it not be better to start out by saying that there is no consensus on what orbifolds “really are” and lay out three points of view: traditional, Moerdijk’s “orbifolds as groupoids” (called “modern” by Adem and Ruan in their book) and orbifolds as stacks?

Urs Schreiber: please, go ahead. It would be appreciated.

end of old query box discussion

]]>Definition of extensional $Pi$-type structure taken from *Natural models of homotopy type theory*

Should we develop how to get application, $\beta$ and $\eta$ here or should we leave it to the interpretation ?

]]>Created categorical model of dependent types, describing the various different ways to strictify category theory to match type theory and their interrelatedness. I wasn’t sure what to name this page — or even whether it should be part of some other page — but I like having all these closely related structures described in the same place.

]]>added graphics of the light hadron masses from Fodor-Hoelbling 12

]]>I fixed a broken link to Guy Moore’s lectures

]]>Added missing axiom. To see that an axiom like that is necessary, just observe that in the former formulation (without the first assumption) the requirements for meet and join were symmetric.

]]>The new second edition is recorded at Practical Foundations for Programming Languages plus a link to a description of the changes.

]]>added pointer to

- R. P. Brent, J. van de Lune, H. J. J. te Riele and D. T. Winter,
*On the Zeros of the Riemann Zeta Function in the Critical Strip. II*, Mathematics of Computation Mathematics of Computation Vol. 39, No. 160 (Oct., 1982), pp. 681-688 (doi:10.2307/2007345 )

for computer-checks of the Riemann hypothesis. (there are probably more recent such?)

]]>Created semi-simplicial set, mainly as a repository for some terminological remarks. I would welcome anyone more knowledgeable about the history to correct or improve it!

]]>Added some remark on the order of a semiring. Actually, does anybody know if any semiring embedds into a semifield?

]]>Created page with some basic definitions.

]]>I think JB used to stay on top of this and delete such comments, but the nCafe currently has 2 questionable recently new entries:

1) This crackpot entry.

2) This entry with the only purpose being to include a url to “cures for candida”.

The 2nd should be deleted and probably the 1st unless crackpots are kept for amusement purposes.

Who has the privileges to delete such these days?

]]>