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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 are special cases of stacks, there is an evident definition of cohomology of orbifolds, given by forming (stable) homotopy groups of derived hom-spaces
into any desired coefficient ∞-stack (or sheaf of spectra) .
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
for a given -action on some manifold , should coincide with the -equivariant cohomology of . However, such an identification (1) is not unique: For any closed subgroup, we have
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 of -equivariant Bredon cohomology is replaced by the global orbit category whose objects are the delooping stacks , and then any orbifold becomes an (∞,1)-presheaf over by the evident “external Yoneda embedding”
More generally, this makes sense for any orbispace. In fact, as a construction of an (∞,1)-presheaf on it makes sense for any ∞-stack, but supposedly precisely if is an orbispace among all ∞-stacks does the cohomology of in the sense of global equivariant homotopy theory coincide the cohomology of 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)
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
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
The new second edition is recorded at Practical Foundations for Programming Languages plus a link to a description of the changes.
added pointer to
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 at cobordism hypothesis a pointer to
where the case for -categories is spelled out and proven in detail.
added a bit more text to the Idea-section at Wick rotation and in particular added cross-links with Osterwalder-Schrader theorem.
created computational trinitarianism, combining a pointer to an exposition by Bon Harper (thanks to David Corfield) with my table logic/category-theory/type-theory.
discovered by chance that we have this stub entry here. Added some lines of an Idea-section and cross-linked with thermal quantum field theory and Wick rotation
I do not understand the entry G-structure. G-structure is, as usual, defined there as the principal -subbundle of the frame bundle which is a -principal bundle. I guess this makes sense for equivariant injections along any Lie group homomorphism . The entry says something about spin structure, warning that the group is not a subgroup of . So what is meant ? The total space of a subbundle is a subspace at least. Does this mean that I consider the frame bundle first as a (non-principal) -bundle by pulling back along a fixed noninjective map and then I restrict to a chosen subspace on which the induced action of Spin group is principal ?