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- Discussion Type
- discussion topicorbifold cohomology
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Urs
- Last Active 1 hour ago

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)

- Discussion Type
- discussion topicmass gap
- Category Latest Changes
- Started by Urs
- Comments 13
- Last comment by Urs
- Last Active 5 hours ago

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)

- Discussion Type
- discussion topicInitiality Project - Type Theory
- Category Latest Changes
- Started by Mike Shulman
- Comments 212
- Last comment by DavidRoberts
- Last Active 11 hours ago

- Discussion Type
- discussion topiclogarithm
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active 18 hours ago

- Discussion Type
- discussion topicinferior limit
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active 20 hours ago

- Discussion Type
- discussion topicInitiality Project - References
- Category Latest Changes
- Started by Alizter
- Comments 8
- Last comment by Mike Shulman
- Last Active 20 hours ago

- Discussion Type
- discussion topicwallpaper group
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicRiemannian orbifold
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicorbifold
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by Urs
- Last Active 1 day ago

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

- Discussion Type
- discussion topicInitiality Project - Semantics - Pi-types
- Category Latest Changes
- Started by kyod
- Comments 8
- Last comment by kyod
- Last Active 1 day ago

- Discussion Type
- discussion topiccategorical model of dependent types
- Category Latest Changes
- Started by Mike Shulman
- Comments 25
- Last comment by kyod
- Last Active 1 day ago

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.

- Discussion Type
- discussion topichadron
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

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

- Discussion Type
- discussion topiclattice gauge theory
- Category Latest Changes
- Started by Tim_Porter
- Comments 3
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topiclattice
- Category Latest Changes
- Started by Daniel Luckhardt
- Comments 3
- Last comment by Todd_Trimble
- Last Active 1 day ago

- Discussion Type
- discussion topicPractical Foundations for Programming Languages
- Category Latest Changes
- Started by David_Corfield
- Comments 9
- Last comment by Mike Shulman
- Last Active 1 day ago

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

- Discussion Type
- discussion topicRiemann hypothesis
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active 2 days ago

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?)

- R. P. Brent, J. van de Lune, H. J. J. te Riele and D. T. Winter,

- Discussion Type
- discussion topicsemi-simplicial set
- Category Latest Changes
- Started by Mike Shulman
- Comments 9
- Last comment by Richard Williamson
- Last Active 2 days ago

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!

- Discussion Type
- discussion topicidempotent semiring
- Category Latest Changes
- Started by Daniel Luckhardt
- Comments 4
- Last comment by DavidRoberts
- Last Active 2 days ago

- Discussion Type
- discussion topicInitiality Project - Raw Syntax
- Category Latest Changes
- Started by Mike Shulman
- Comments 56
- Last comment by atmacen
- Last Active 2 days ago

- Discussion Type
- discussion topicproton spin crisis
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 2 days ago

- Discussion Type
- discussion topiccobordism hypothesis
- Category Latest Changes
- Started by Urs
- Comments 27
- Last comment by Urs
- Last Active 2 days ago

added at cobordism hypothesis a pointer to

- Yonatan Harpaz,
*The Cobordism Hypothesis in Dimension 1*(arXiv:1210.0229)

where the case for $(\infty,1)$-categories is spelled out and proven in detail.

- Yonatan Harpaz,

- Discussion Type
- discussion topicFulton-MacPherson operad
- Category Latest Changes
- Started by Urs
- Comments 19
- Last comment by Todd_Trimble
- Last Active 3 days ago

- Discussion Type
- discussion topiclanguage
- Category Latest Changes
- Started by Daniel Luckhardt
- Comments 2
- Last comment by Urs
- Last Active 3 days ago

- Discussion Type
- discussion topicWick rotation
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active 3 days ago

added a bit more text to the Idea-section at

*Wick rotation*and in particular added cross-links with*Osterwalder-Schrader theorem*.

- Discussion Type
- discussion topicn-point function
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 3 days ago

- Discussion Type
- discussion topiccomputational trinitarianism
- Category Latest Changes
- Started by Urs
- Comments 28
- Last comment by Urs
- Last Active 3 days ago

created

*computational trinitarianism*, combining a pointer to an exposition by Bon Harper (thanks to David Corfield) with my table logic/category-theory/type-theory.

- Discussion Type
- discussion topicInitiality Project - Raw Syntax - Pi-types
- Category Latest Changes
- Started by Mike Shulman
- Comments 5
- Last comment by Mike Shulman
- Last Active 4 days ago

- Discussion Type
- discussion topicarctangent
- Category Latest Changes
- Started by TobyBartels
- Comments 1
- Last comment by TobyBartels
- Last Active 4 days ago

- Discussion Type
- discussion topicKMS state
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 4 days ago

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*

- Discussion Type
- discussion topicG-structure
- Category Latest Changes
- Started by zskoda
- Comments 14
- Last comment by Urs
- Last Active 5 days ago

I do not understand the entry G-structure. G-structure is, as usual, defined there as the principal $G$-subbundle of the frame bundle which is a $GL(n)$-principal bundle. I guess this makes sense for equivariant injections along any Lie group homomorphism $G\to GL(n)$. The entry says something about spin structure, warning that the group $Spin(n)$ is not a subgroup of $GL(n)$. 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) $Spin(n)$-bundle by pulling back along a fixed noninjective map $Spin(n)\to GL(n)$ and then I restrict to a chosen subspace on which the induced action of Spin group is principal ?