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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeAug 31st 2018
    • (edited Aug 31st 2018)

    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 𝒢\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 (𝒢,E)π H(𝒢,E) H^\bullet(\mathcal{G}, E) \;\coloneqq\; \pi_\bullet \mathbf{H}( \mathcal{G}, E )

    into any desired coefficient ∞-stack (or sheaf of spectra) EE.

    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

    𝒢XGAAAA(1) \mathcal{G} \;\simeq\; X \sslash G \phantom{AAAA} (1)

    for a given GG-action on some manifold XX, should coincide with the GG-equivariant cohomology of XX. However, such an identification (1) is not unique: For GKG \subset K any closed subgroup, we have

    XG(X× GK)K. 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 GOrb_G of GG-equivariant Bredon cohomology is replaced by the global orbit category Orb glbOrb_{glb} whose objects are the delooping stacks BG*G\mathbf{B}G \coloneqq \ast\sslash G, and then any orbifold 𝒢\mathcal{G} becomes an (∞,1)-presheaf y𝒢y \mathcal{G} over Orb glbOrb_{glb} by the evident “external Yoneda embedding

    y𝒢H(BG,𝒢). y \mathcal{G} \;\coloneqq\; \mathbf{H}( \mathbf{B}G, \mathcal{G} ) \,.

    More generally, this makes sense for 𝒢\mathcal{G} any orbispace. In fact, as a construction of an (∞,1)-presheaf on Orb glbOrb_{glb} it makes sense for 𝒢\mathcal{G} any ∞-stack, but supposedly precisely if 𝒢\mathcal{G} is an orbispace among all ∞-stacks does the cohomology of y𝒢y \mathcal{G} in the sense of global equivariant homotopy theory coincide the cohomology of 𝒢\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)

    diff, v2, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeSep 2nd 2018
    • (edited Sep 2nd 2018)

    Eventually I’d like to get a better idea of the following:

    Given an orbifold, regarded as a global equivariant homotopy type as indicated on p. ix-x of arXiv:1802.09382, and given a cocycle on it with coefficients in something like the global equivariant sphere spectrum; how is this cocyle characterized in terms of an atlas by charts R nG iR^n \sslash G_i ?

    Does the cocycle on the orbifold restrict to a G iG_i-equivariant Bredon cocycle on the iith chart? If so, how are these restrictions compatible along gluings of orbifold charts?

    This ought to be straightforward to answer by unwinding the definitions. But I still need to think through this. Is there any discussion of this kind of thing in the literature?

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 3rd 2018

    Peter May writes 7 years ago

    I don’t know of any connection between orbifolds and global spectra,

    so I guess anything in this area is of very recent date. Perhaps ask at MO.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2018
    • (edited Sep 3rd 2018)

    Just realized that for plain orbifolds (not differential- or supergeometric etc.) my question seems to trivialize via Charles Rezks’s observation that global homotopy theory is cohesive.

    I wrote:

    Given an orbifold, regarded as a global equivariant homotopy type as indicated on p. ix-x of arXiv:1802.09382, and given a cocycle on it with coefficients in something like the global equivariant sphere spectrum; how is this cocyle characterized in terms of an atlas by charts R nG iR^n \sslash G_i ?

    But the global sphere spectrum is the image of the plain sphere spectrum under the left adjoint from stable homotopy theory to global stable homotopy theory (e.g. p. 6 of arXiv:1802.09382), which I suppose is the direct stabilization of the left adjoint from homotopy theory to global homotopy theory, which Charles observed has a further left adjoint given by 1-categorical quotients (5.1 on p. 15 of Global Homotopy Theory and Cohesion.)

    This would mean that, in particular, global equivariant cohomotopy of an orbifold is just plain cohomotopy of its quotient space.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2018
    • (edited Sep 3rd 2018)

    David, the relation of global homotpy theory to orbispaces (hence in particular orbifolds) is the content of

    [The key theorem here is that by Henriques-Gepner in their article on orbispaces.]

    While I appreciate this, at present this goes through too many Quillen equivalences (plus one non-Quillen equivalence) for me to have a good feeling for what actually happens when regarding an orbifold in global homotopy theory.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2018

    Oh, I see my conceptual mistake in #4: Since locally I want to have equivariant cohomology, I need to use the “faithful” version of orbispaces, i.e. version (2) in Henriques-Gepner, equivalently what Charles Rezk just calls orbispaces, as opposed to global spaces. I’ll follow Charles and just say “orbispace” for this now.

    So I guess specifically for the case of ADE-equivariant cohomotopy, I want to do this:

    a) Regard S 4S^4 with given SU(2)SU(2)-action as an orbispace, in the above sense. b) Regard an orbifold 𝒳\mathcal{X} with isotropy groups being finite subgroups of SU(2)SU(2) as an orbispace. Then 3) consider maps of orbispaces from the latter to the former as the relevant global equivariant cohomotopy cocyles of the orbifold:

    GlobalEquivariantCohomotopy(𝒳)π 0Maps Orbispaces(𝒳,S 4SU(2))π 0Maps GlobalSpaces/𝒩(𝒳,S 4SU(2)). GlobalEquivariantCohomotopy(\mathcal{X}) \;\coloneqq\; \pi_0 Maps_{Orbispaces}( \mathcal{X}, S^4\sslash SU(2) ) \;\simeq\; \pi_0 Maps_{GlobalSpaces/\mathcal{N}}( \mathcal{X}, S^4\sslash SU(2) ) \,.

    where the equivalence is by Charles’s result in Global Homotopy Theory and Cohesion (e.g. top of p. 4).

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 3rd 2018

    What is 𝒩\mathcal{N} here?

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 3rd 2018

    Where’s a good spot to note that 𝒩\mathcal{N} is the ’normal subgroup classifier’ (Rezk 14, 4.1)?

    Did you ever get to the bottom of that sense you had that global cohesion is a little different from other forms?:

    one reason why it may be hard to match the intended intuition for cohesion to the cohesion you find is that, to my mind at least, the cohesion you find is curiously “shifted to the left” in an unexpected way.

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 3rd 2018

    Ah, I see: 𝒩\mathcal{N} is the terminal orbispace.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2018
    • (edited Sep 3rd 2018)

    Yes, 𝒩\mathcal{N} is the group object in \infty-presheaves over the full subcategory on BG\mathbf{B}G-s inside all smooth \infty-stacks which is such that a faithful morphism into it characterizes its domain as an “orbispace” among the “global equivariant spaces”. This is the way how Charles gives intrinsic meaning (in Global Homotopy Theory and Cohesion) to othe distinction between cases (1) and (2) in the original Henriques-Gepner 07.

    The way to think about it is (in my words) that a map between stacks alone is not necessarily an “equivariant map” in any way, but it is GG-equivariant for given GG if it lifts to a map in the slice over BG\mathbf{B}G (by the discussion at infinity-action). To make this work for all GG at once, we need a context where there exists a “universal compact Lie group”. This is really what global equivariant homotopy theory is about. Here 𝒩\mathcal{N} is the “delooping of that universal compact Lie group”.

    I am thinking that it will be good to develop intuition on the simpler case where we have a more restricted family \mathcal{F} of admissible isotropy groups so that the “universal” one may turn out to be an actual group. This is just what happens, in particular, in the ADE-equivariant situation:

    Here \mathcal{F} is the set of all finite subgroups of SU(2)SU(2), and SU(2)SU(2) is the universal among these, in the suitable generalized sense. Concretely this comes down to the elementary statement that if 𝒳\mathcal{X} is an orbifold with isotropy groups being finite subgroups of SU(2)\mathrm{SU}(2), then there is a canonical faithful morphism 𝒳BSU(2)\mathcal{X} \to \mathbf{B} SU(2), and for AA any space with an SU(2)SU(2)-action, a global SU(2)SU(2)-equivariant cocyle on 𝒳\mathcal{X} is simply a map of stacks 𝒳ABSU(2)\mathcal{X} \longrightarrow A\sslash \mathbf{B}SU(2) in the slice over BSU(2)\mathbf{B}SU(2).

    What then does global equivariant homotopy theory do for us, if it is that simple? Answer: It provides (as advertized in Henriques-Gepner 07) the analog of Elmendorf’s theorem, which allows to express such maps of stacks equivalently as systems of maps on fixed point strata.

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 3rd 2018

    Does \mathcal{F} have to satisfy any particular properties? I recall trying to see if the finite subgroups of SU(2)SU(2) formed a global family in the sense that Schwede uses. It seems that past me thought they formed a global family, but not a multiplicative global family.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeSep 3rd 2018

    Hi David R., thanks for highlighting this technical issue again, but here I just meant to illustrate the role of 𝒩\mathcal{N}. By the nature of 𝒩\mathcal{N} we may slice over 𝒩\mathcal{N} instead of over BSU(2)\mathbf{B}SU(2), and if an orbifold only happens to have isotropy in finite ADE-groups, the concept of orbifold cohomology won’t change; it’s embedded inside the larger theory of more general orbispaces.

    • CommentRowNumber13.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 3rd 2018
    • (edited Sep 3rd 2018)

    Ok, good point.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2018
    • (edited Sep 9th 2018)

    I have tried to make this point in #12 more explicit, by adding to the Idea-section the following paragraphs (here):

    We may make this more explicit in the case where one considers the class of orbifolds 𝒳\mathcal{X} whose isotropy groups are any finite subgroup GιG globG \overset{\iota}{\hookrightarrow} G_{glob} of a fixed compact Lie group G globG_{glob}. Such orbifolds carry canonical morphisms to the delooping stack BG glob\mathbf{B}G_{glob}, which are locally, for global quotients 𝒳XG\mathcal{X} \simeq X \sslash G, given by

    ρ:XG(X*)GBGBιBG glob. \rho \;\colon\; X \sslash G \overset{ (X \to \ast) \sslash G }{\longrightarrow} \mathbf{B}G \overset{ \mathbf{B} \iota }{\longrightarrow} \mathbf{B}G_{glob} \,.

    By the general discussion at ∞-action any such morphism exhibits an ∞-action of G globG_{glob} on the homotopy fiber hofib(ρ)hofib(\rho) of ρ\rho, together with an equivalence

    (1)AAhofib(ρ)G globXG. (1) \phantom{AA} hofib(\rho)\sslash G_{glob} \;\simeq\; X \sslash G \,.

    But that homotopy fiber is directly computed to be

    hofib(ρ) *× BG glob(BG glob) Δ[1]× BG globBG (X×G glob)/G X× GG glob, \begin{aligned} hofib(\rho) & \simeq \ast \times_{\mathbf{B}G_{glob}} \big( \mathbf{B}G_{glob} \big)^{\Delta[1]} \times_{\mathbf{B}G_{glob}} \mathbf{B}G \\ & \simeq \big( X \times G_{glob} \big) / G \\ & \simeq X \times_G G_{glob} \,, \end{aligned}

    where in the first step we used the factorization lemma, and the remaining steps follow by direct inspection. Plugging this back into (1) yields the equivalence (X× GK)KXG( X \times_G K ) \sslash K \simeq X \sslash G for GKG globG \hookrightarrow K \coloneqq G_{glob}.

    In conclusion, if 𝒳\mathcal{X} is any orbifold, i.e. not necessarily a global homotopy quotient X iG iX_i \sslash G_i, but locally of this form for each G iG_i some finite subgroups of G globG_{glob}, then it comes with a canonical morphism of topological stacks 𝒳ρBG glob\mathcal{X} \overset{\rho}{\to} \mathbf{B}G_{glob}, and so for 𝒜BG glob\mathcal{A} \overset{}{\to} \mathbf{B}G_{glob} any coefficient ∞-stack in the slice over BG glob\mathbf{B}G_{glob}, we may take the global equivariant orbifold cohomology to be given by homotopy classes of morphisms in the slice:

    H glob(𝒳,𝒜)π 0H /BG glob(𝒳,𝒜). H_{glob}(\mathcal{X}, \mathcal{A} ) \;\simeq\; \pi_0 \mathbf{H}_{/ \mathbf{B}G_{glob}} \big( \mathcal{X}, \mathcal{A} \big) \,.

    Cocycles in this “global equivariant” cohomology are then such that on each chart of the form U iG iU_i \sslash G_i they restrict to cocycles in G iG_i-equivariant cohomology of U iU_i, in a way that is compatible with the above re-identifications (eq:EquivalenceOfOrbifolds).

    diff, v4, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2018
    • (edited Sep 9th 2018)

    added further paragraphs to make the connection to the “Gepner-Henriques global Elmendorf theorem”:

    Notice that if also the coefficient 𝒜BG glob\mathcal{A} \overset{}{\to} \mathbf{B} G_{glob} is faithful (0-truncated) as an object in the slice, then, by the orthogonality of the (n-connected, n-truncated) factorization system for n=0n = 0, there is a contractible space of homotopies ϕ\phi in the data for a cocycle

    𝒳 AAcAA 𝒜 ρ ϕ f BG globAAA=AAA𝒳 AcA 𝒜 = ! f 𝒳 AρA BG glob \array{ \mathcal{X} && \overset{ \phantom{AA} c \phantom{AA} }{\longrightarrow} && \mathcal{A} \\ & {}_{\mathllap{\rho}}\searrow &\swArrow_{\phi}& \swarrow_{\mathrlap{f}} \\ && \mathbf{B}G_{glob} } \phantom{AAA} = \phantom{AAA} \array{ \mathcal{X} &\overset{ \phantom{A} c \phantom{A} }{\longrightarrow}& \mathcal{A} \\ {}^{=}\big\downarrow & \swArrow_{\exists !} & \big\downarrow{}^{\mathrlap{f}} \\ \mathcal{X} &\underset{ \phantom{A} \rho \phantom{A} }{\longrightarrow}& \mathbf{B}G_{glob} }

    Moreover, in this case the cocyle morphism cc itself is necessarily faithful (0-truncated). This means that the full sub-(∞,1)-category of the slice (∞,1)-category on the fatihful/0-truncated morphisms is equivalently the non-full sub \infty-category of the corresponding domain ∞-stacks, but with fatihful/0-truncated morphisms between them:

    H /BG glob(𝒳,𝒜)H faith(𝒳,𝒜), \mathbf{H}_{/\mathbf{B}G_{glob}} \big( \mathcal{X}, \mathcal{A} \big) \;\simeq\; \mathbf{H}^{faith} \big( \mathcal{X}, \mathcal{A} \big) \,,

    which hence gives an equivalent description of the global equivariant orbifold cohomology in (eq:CocyclesInTheSlice).

    This perspective paves the way to the equivalent description in terms of systems of fixed point loci:

    and then

    In global equivariant homotopy theory the plain orbit category Orb GOrb_G used in GG-equivariant Bredon cohomology is replaced by the global orbit category Orb glbOrb_{glb} whose objects are the delooping stacks BG*G\mathbf{B}G \coloneqq \ast\sslash G and whose morphisms are the faithful/0-truncated morphisms between these (Henriques-Gepner 07, 4.1, Rezk 14, 4.5). Then any stack 𝒳\mathcal{X} (orbifold, orbispace) becomes an (∞,1)-presheaf y𝒳y \mathcal{X} over Orb glbOrb_{glb} by the evident “external Yoneda embedding

    y𝒳H faith(BG,𝒳). y \mathcal{X} \;\coloneqq\; \mathbf{H}^{faith}( \mathbf{B}G, \mathcal{X} ) \,.

    The generalization of Elmendorf’s theorem to global equivariant homotopy theory, hence to the application of orbifold cohomology, is now the statement that this construction induces equivalences of cocycle ∞-groupoids

    H faith(𝒳,𝒜)PSh (Orb glb)(y𝒳,y𝒳). \mathbf{H}^{faith} \big( \mathcal{X}, \mathcal{A} \big) \;\simeq\; PSh_{\infty}(Orb_{glb}) ( y \mathcal{X}, y \mathcal{X} ) \,.

    This is the staement of Henriques-Gepner 07, main theorem (4) on p. 5 in version (2) according to p. 8. With particular emphasis on its application to orbifold cohomology, this is highlighted in (Schwede 17, Introduction, Schwede 18, p. ix-x). See also Rezk 14, section 4.

    In summary, the definition of global equivariant orbifold cohomology according to (eq:CocyclesInTheSlice) is equivalent, via (eq:Faithful) and (eq:GlobalElmendorfTheorem), to

    H(𝒳,𝒜)π 0PSh (Orb glb)(y𝒳,y𝒶). H(\mathcal{X}, \mathcal{A}) \;\simeq\; \pi_0 PSh_\infty(Orb_{glb})( y\mathcal{X}, y \mathcal{a} ) \,.

    diff, v5, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2018

    dropped a comment also in the comment section here alongside Felix’s exposition “Differential Geometry in Modal HoTT

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeNov 14th 2018
    • (edited Nov 14th 2018)

    fixed a typo in the first line of the computation of the homotopy fiber (here) and added a cartoon picture of what the computation does

    diff, v8, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2018
    • (edited Nov 23rd 2018)

    I will rewrite that discussion again.

    The point is really that we may regard any smooth \infty-groupoid 𝒳HSh (CartSp)\mathcal{X} \in \mathbf{H} \coloneqq Sh_\infty(CartSp) as a presheaf eq(𝒳)eq(\mathcal{X}) of GlobalOrbits={finite groupoidsBKwith faithful functors}GlobalOrbits = \{\text{finite groupoids} B K \text{with faithful functors}\} by

    BK[Disc(BK),𝒳] H B K \mapsto [Disc(B K), \mathcal{X}]_{\mathbf{H}}

    and if 𝒳\mathcal{X} happens to be an orbifold, then this is its correct incarnation as a global equivariant thing. In the sense that for GG any finite group, AGSpaceA \in G Space any GG-space, regarded as

    Discδ GAPSh (GlobalOrbits,H) Disc \delta_G A \;\in\; PSh_\infty( GlobalOrbits, \mathbf{H} )

    the correct orbifold cohomology is

    PSh (GlobalOrbits,H)(e(𝒳),Discδ GA). PSh_\infty( GlobalOrbits, \mathbf{H} )( e(\mathcal{X}), Disc \delta_G A ) \,.

    and this expression will automatically first regard 𝒳\mathcal{X} as a GG-space, if possible.

    • CommentRowNumber19.
    • CommentAuthorGuest
    • CommentTimeNov 29th 2018

    Ah, and that assignment in the first display of #18 is of course the sharp modality with respect to the global equivariant structure.

    So now I finally see the neat picture of orbifold cohomology, I think:


    Orbifold cohomology

    Let H\mathbf{H} be a differentially cohesive \infty-topos. Write

    SingularitiesGroupoids 1,fin cn Singularities \;\coloneqq\; Groupoids_{1,fin}^{cn}

    for the (2,1)(2,1)-category of finite connected groupoids and, following Charles Rezk, write

    𝔹GSingularities \mathbb{B}G \;\in\; Singularities

    for the object given by the delooping groupoid on some finite group GG.

    The notation is to distinguish from the delooping groupoid BGH\mathbf{B}G \in \mathbf{H}, and that distinction is the all-important distinction that now drives the theory: We are saying that an orbifold singularity is not really a geometric groupoid BG=*GH\mathbf{B}G =\ast \sslash G \in \mathbf{H}, as it would be in the traditional spirit of Moerdijk-Pronk. Rather, that is just one aspect of it, the other, dual aspect being the genuine point aspect */G=*\ast / G = \ast, which is what physicists really think of when they speak of orbifolds, thinking of them really as manifolds with singular points, but not with huge infinite dimensional spaces such as B 2=P \mathbf{B} \mathbb{Z}_2 = \mathbb{R}P^\infty attached to the points. Instead, the true notion of orbifold singularity is the unification/resolution of these two opposing aspects, as formalized by the cohesive adjoint modalities:


    H singSh (Singularities,H) \mathbf{H}_{sing} \;\coloneqq\; Sh_\infty( Singularities, \mathbf{H} )

    Remark on terminology. The category SingularitiesSingularities is usually called “OrbOrb version 1” (Gepner-Henriques) or “GLobGLob (Rezk)” or just “OrbOrb” (Koerschgen, Schwede), but let’s give it a name that actually reflects its meaning (and I found this the hard part of figuring out what’s really going on :-): The objects in this (2,1)(2,1)-category are the local models for orbifold singularities. In particular if SynthCartSpaceSynthCartSpace is a site of definition of H\mathbf{H} with objects such as n\mathbb{R}^n, then a site of definition of H sing\mathbf{H}_{sing} is Singularities×SynthCartSpaceSingularities \times SynthCartSpace with objects such as n×𝔹G\mathbb{R}^n \times \mathbb{B}G, which are manifestly the local model spaces for orbifold singularities.

    Since SingularitiesSingularities is a small \infty-site with finite products, we have immediately that H sing\mathbf{H}_{sing} is cohesive over H\mathbf{H}, as in section 5.1 of Charles Rezk’s Global Homotopy Theory and Cohesion. I’ll denote the corresponding modalities with subscript sing{}_{sing}.


    For VHDisc singH singV \in \mathbf{H} \overset{Disc_{sing}}{\hookrightarrow} \mathbf{H}_{sing} a group object, we may consider V-manifolds XHDisc singH singX \in \mathbf{H} \overset{Disc_{sing}}{\to} \mathbf{H}_{sing}.

    For GG a finite group, say that such a VV-manifold is an orbifold with singularities/isotropy groups in GG if there is a 0-truncated (i.e. “faithful”) morphism

    (XBG)HDisc singH sing (X \longrightarrow \mathbf{B}G) \;\in\; \mathbf{H} \overset{Disc_{sing}}{\hookrightarrow} \mathbf{H}_{sing}

    Picking any such is a real choice, I suppose. But let’s pick one and regard the result as

    𝒳H /BGDisc sing(H sing) /BG \mathcal{X} \;\in\; \mathbf{H}_{/\mathbf{B}G} \overset{Disc_{sing}}{\hookrightarrow} \left( \mathbf{H}_{sing}\right)_{/\mathbf{B}G}

    This is our orbifold regarded as a geometric groupoid, in Moerdijk-Pronk-spirit.

    The first point now is that this is not the right incarnation of orbifolds, for purposes of orbifold cohomology: The cohomology in the slice over BG\mathbf{B}G is “geometric cohomology with local coefficients”, but it is not geometric Bredon equivariant cohomology.

    The second point now is that there is a neat way to improve on this: The right incarnation of the orbifold is the sharp-modal aspect, with respect to the singularity-cohesion:

    sing𝒳 =coDisc singΓ singDisc sing(XBG) coDisc sing(XBG) =coDisc sing(X)𝔹G \begin{aligned} \sharp_{sing} \mathcal{X} & = coDisc_{sing} \Gamma_{sing} Disc_{sing} (X \to \mathbf{B}G) \\ & \simeq coDisc_{sing} (X \to \mathbf{B}G) \\ & =coDisc_{sing}(X) \to \mathbb{B}G \end{aligned}

    To see what the last line is, use that

    Γ sing(𝔹G)=Singularities(𝔹1,𝔹G)=BG \Gamma_{sing}(\mathbb{B}G) \;=\; Singularities( \mathbb{B}1 , \mathbb{B}G) \;=\; \mathbf{B}G

    and the adjunction (Γ singcoDisc sing)(\Gamma_{sing} \dashv coDisc_{sing}):

    coDisc sing(X): n×𝔹GH( n,×BG) coDisc_{sing}(X) \;\colon\; \mathbb{R}^n \times \mathbb{B}G \;\mapsto\; \mathbf{H}( \mathbb{R}^n, \times \mathbf{B}G )

    In words: coDisc singcoDisc_{sing} reads in an orbifold regarded as a smooth groupoid, and then replaces all its BG\mathbf{B}G-singularities with the refined 𝔹G\mathbb{B}G-singularties.

    In particular coDisc sing(BG)𝔹GcoDisc_{sing}(\mathbf{B}G) \simeq \mathbb{B}G.

    Similarly it is immediate that with XBGX \to \mathbf{B}G being faithful (0-truncated), so is coDisc sing(X)𝔹GcoDisc_{sing}(X) \to \mathbb{B}G.

    But this means, by section 4 in Global Homotopy Theory and Cohesion, that our improved orbifold sing(𝒳)\sharp_{sing}(\mathcal{X}) with singularities in GG is indeed an object now in geometric GG-equivariant homotopy theory (“geometric” because we are over the differentially cohesive base \infty-topos H\mathbf{H} instead of just over Groupoids Groupoids_\infty.)

    In particular now for

    𝒜in((H sing) /𝔹G) faith \mathcal{A} \;in\; \big( (\mathbf{H}_{sing})_{/\mathbb{B}G} \big)_{faith}

    any coefficient object, the correct Bredon-equivariant geometric/differential cohomology of 𝒳\mathcal{X} is

    H G(𝒳,𝒜)π 0H sing( sing𝒳,𝒜). H_{G}(\mathcal{X}, \mathcal{A}) \;\coloneqq\; \pi_0 \mathbf{H}_{sing}\left( \sharp_{sing} \mathcal{X}, \mathcal{A}\right) \,.


    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeNov 29th 2018

    Somehow I ended up sending this as “Guest”. Which means I can’t fix now the two typos in the last two displays. Anyway, I’ll write this out into the entry, eventually.

    • CommentRowNumber21.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 29th 2018

    Great to see the global equivariance modalities being used! Did you resolve in your mind what you took to be a “shift to the left” of the adjoint quadruple back here?

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeNov 29th 2018
    • (edited Nov 29th 2018)

    Back then my intuition was no good. Took me until now to get a clear intuition for this whole business. Part of the problem is, I dare say, that much of the established terminology is somewhere between unhelpful and misleading. But after a little renaming (“GlobGlob\rightsquigarrowSingularitiesSingularities”) Charles’ cohesion is of course the way to see the light.

    Here is what I now think is the right intuition:

    Envision the picture of an orbifold singularity (say the teardrop, for definiteness) and hold a magic magnifying glass over the singular point. Inside the magnifying glass you see resolved the singular point as a fuzzy fattened point, labeled 𝔹G\mathbb{B}G.

    Removing the magnifying glass, what one sees with the bare eye depends on how one squints:

    1. The physicist says that what he sees is a singular point, but a point after all. That’s *=*/G\ast = \ast / G.

    2. The Lie geometer says that what he sees is a point with a trivial GG action, a groupoid BG=*G\mathbf{B}G =\ast \sslash G.

    These two aspects are two opposite extreme aspects of the orbifold singularity 𝔹G\mathbb{B}G, but the orbifold singularity is more than both of these aspects. The real nature of an orbifold singularity is really a point, not a big classifying space, but it also does remember the group action, for that characterizes how the singularity is being singular.

    This state of affairs is exactly matched by the cohesive adjoint modalities:

    orbifold singularity𝔹G ʃ sing Aopposite extremeaspects of orbifold singularity sing plain quotient*=*/G homotopy quotientBG=*G \array{ && { \text{orbifold singularity} \atop {\mathbb{B}G} } \\ & {}^{\mathllap{ʃ_{sing}}}\swarrow & {{\phantom{A}} \atop { \text{opposite extreme} \atop \text{aspects of orbifold singularity} }} & \searrow^{\mathrlap{ \flat_{sing} }} \\ { \text{plain quotient} \atop {\ast = \ast/G} } && && { \text{homotopy quotient} \atop { \mathbf{B}G = \ast \sslash G } } }
    • CommentRowNumber23.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 29th 2018
    • (edited Nov 29th 2018)

    So we have points-to-pieces, sing𝒳𝒳ʃ sing𝒳\flat_{sing} \mathcal{X} \to \mathcal{X} \to ʃ_{sing} \mathcal{X}. In the teardrop case, this is supposed to map the big classifying space to the fuzzy fattened point to the singular point.

    What would it be for pieces to have points?

    Presumably your SingularitiesSingularities could be any global family.

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeNov 29th 2018
    • (edited Nov 29th 2018)

    What would it be for pieces to have points?

    Good question. We will need to use the \infty-categorical refinement of “pieces have points” and ask whether the comprison map is nn-connected for various nn.

    So it is always (1)(-1)-connected, namely an effective epi, hence verifies “pieces have points” in the underlying 1-topos.

    The comparison map will also always be 0-connected.

    It will be 1-connected precisely away from the orbifold singularities.

    Presumably your SingularitiesSingularities could be any global family.

    I am having some debate about this with Vincent. My point of view is (until being convinced otherwise, which I am open to): Orbifolds are all about finite isotropy groups and SingularitiesSingularities should hence be finite connected groupoids and nothing else. Also the nice story in #19 with refined orbifolds being coDisc singcoDisc_{sing} of their Lie groupoid incarnation only works for discrete groups.

    • CommentRowNumber25.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 29th 2018

    Another natural thing to wonder about is the differential cohomological diagram in this case. What kind of fracturing goes on for stable singular spaces?

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeNov 29th 2018
    • (edited Nov 29th 2018)

    Since we are talking about \infty-presheaves on the “global orbit category, version 1”, the relation to global equivariant stable homotopy theory is manifestly built in. For most people concerned here, this is the main motivation for considering this setup in the first place.

    What seems to have remained open is really just that definition of what exactly, given some globally equivariant spectrum (or better: sheaf of spectra), it should really mean to take the cohomology of an orbifold with these coefficients.

    One thought that kept misleading me for a long while is that, due to the suggestive name “global equivariant”, we should somehow get an equivariant cohomology of orbifolds without ever slicing over anything. But this, I came to realize, is just the wrong idea, and this slicing will be necessary, meaning equivalently that we will always be considering a twisted kind of global equivariant stable cohomology.

    Or so I came to think. Experts should please disagree.

    • CommentRowNumber27.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 29th 2018

    Very nice! (Unfortunately I have no comment at present of of more substance)

    • CommentRowNumber28.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 30th 2018

    “Glob” in Rezk 14, 2.2

    Actually he calls it “Glo”. You included a discussion at global equivariant homotopy theory.

    Also, won’t it be worth pointing out that GloGlo there is an enriched category whose objects are compact Lie groups, so include a word on your choice of SingularitiesSingularities.

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeNov 30th 2018

    I have now typed out details of the picture of orbifold cohomology that I indicated in #19.

    The maths should all be there, but – apart from an fair bit of an Idea-section – I haven’t added much guiding text yet alongside. Also I haven’t proof-read yet for typos etc.

    diff, v22, current

    • CommentRowNumber30.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 1st 2018

    Did some proof-reading.

    diff, v23, current

    • CommentRowNumber31.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2018


    • CommentRowNumber32.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2018

    added guiding lead-in paragraphs to each of the subsections.

    diff, v24, current

    • CommentRowNumber33.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 1st 2018

    Will anything interesting happen as you refine the H\mathbf{H} to a super differential cohesive version? There must be ’super-orbifolds’.

    Some google hits for that, but not as many as might be expected.

    • CommentRowNumber34.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2018

    That’s the very reason why I am looking into this. We may now formally say

    “11d-supergravity enhanced by M-brane charge quantization in supertorsion-free ADE-equivariant differential Cohomotopy in RO-degree \mathbb{H} on super-orbifold spacetimes”.

    This is meant to be interesting, yes. We have now a list of half a dozen or so limiting cases of the space of observables on this enhanced sugra field space, comparing to known classifications of (fractional) brane charges as well as to caloron correlators in non-perturbative 4d field theory, cosmic Galois group action included, and it looks promising.

    • CommentRowNumber35.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 1st 2018

    Look forward to seeing it all written out.

    • CommentRowNumber36.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2018
    • (edited Dec 2nd 2018)

    What might be good symbols for the three modalities of global homotopy theory cohesive over plain homotopy theory? In the entry I have been using ʃ sing sing singʃ_{sing} \dashv \flat_{sing} \dashv \sharp_{sing}, but there must be better choices.

    It might be good if the symbols used could suggest the idea that ʃ singʃ_{sing} produces something that is “too small” or “too rigid”, while sing\flat_{sing} produces something that is “too big” or “too fuzzy”, compared to the real thing.

    • CommentRowNumber37.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 2nd 2018
    • (edited Dec 2nd 2018)

    I guess one advantage of the current way is that H sing\mathbf{H}_{sing} itself is cohesive over Grpd\infty Grpd if H\mathbf{H} is, so that a composition such as ΓΓ sing\Gamma \cdot \Gamma_{sing} is just another form of Γ\Gamma.

    In the space of all forms of cohesion why single out factors of some factorisations as requiring special symbols? But then why differentiate corresponding (co)monads?

    On the other hand, I can see that they’re rather different here.

    • CommentRowNumber38.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2018
    • (edited Dec 2nd 2018)

    so that a composition such as ΓΓ sing\Gamma \cdot \Gamma_{sing} is just another form of Γ\Gamma.

    Ah, I am not looking for alternatives to the symbols ΠDiscΓcoDisc\Pi \dashv Disc \dashv \Gamma \dashv coDisc, but for alternative to the symbols for the corresponding modal operators ʃʃ \dashv \flat \dashv \sharp

    Maybe the symbol


    could be used instead of sing\sharp_{sing}, suggesting a singular tip (in LaTeX I wold maybe rotate it 90 degrees), while


    could then be used for sing\flat_{sing}, suggesting smoothness in the sense of absence of singularities

    • CommentRowNumber39.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2018

    Maybe like so:

    < \lt \;\dashv\; \subset \;\dashv\; \prec

    On the left we see an orbifold where the singularity is just the plain singular quotient.

    In the middle we see an orbifold that has no singularities, it is “smooth” (in the algebro-geometers sense of smooth = no singularities).

    On the right we see the “actual” orbifold singularity.

    • CommentRowNumber40.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2018
    • (edited Dec 2nd 2018)

    Fun what we learn philosophically:

    The duality of dualities

    < \array{ \lt &\dashv& \subset \\ \bot && \bot \\ \subset &\dashv& \prec }

    says that the quality “non-singular” (\subset) is the opposite extreme of two different kinds of singular qualities: on the one hand of “bad quotients” (<\lt) on the other hand of orbifold singularities (\prec). Moreover, these two ways of being opposite extremes to “non-singular” are opposite extremes of being such opposite extremes, and it is in this second-order-duality sense that the concept of bad (naive) quotients is the opposite of the concept of orbifold-singularities.

    • CommentRowNumber41.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 2nd 2018

    Odd that something “bad” should appear? I mean can’t one tell a tale of its inevitable appearance? Or is all downhill as we extend from \prec?

    Could this duality of dualities partake in any larger diagram? Could be the Aufhebung of some other opposition?

    • CommentRowNumber42.
    • CommentAuthorUrs
    • CommentTimeDec 2nd 2018

    That’s a term that came to mind. People say that, “bad quotient”. But let’s find better terminology! If you have suggestions, let me know.

    • CommentRowNumber43.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 2nd 2018

    “Naïve quotient”?

    • CommentRowNumber44.
    • CommentAuthorUrs
    • CommentTimeDec 3rd 2018

    “Naïve quotient”?

    So instead of the object, we blame the mathematician? ;-)

    A more technical term would really be “non-smooth” in the algebro-geometers sense.

    I used to avoid this usage of “smooth” = “non-singular”, since, while of course related, it doesn’t really match the difference between a smooth and a C kC^k-manifold, say.

    But in the grand scheme of things the latter meaning is really absorbed in “cohesive”, and so maybe we should be free to use “smooth” now for “non-singular”.

    Then the pronounciation of the equivariant modalities could now be

    < non-smooth smooth singular\subset & \array{ \lt &\dashv& \subset &\dashv& \prec \\ \text{non-smooth} && \text{smooth} && \text{singular} }


    < singular smooth cosingular \array{ \lt &\dashv& \subset &\dashv& \prec \\ \text{singular} && \text{smooth} && \text{cosingular} }

    or something like this.

    • CommentRowNumber45.
    • CommentAuthorDavidRoberts
    • CommentTimeDec 3rd 2018

    How about “orbits”? Or are you looking for an adjective?

    • CommentRowNumber46.
    • CommentAuthorUrs
    • CommentTimeDec 3rd 2018

    But “orbits” means something else, I’d think. Maybe “orbifold-singular”?

    < singular smooth orbi-singular \array{ \lt &\dashv& \subset &\dashv& \prec \\ \text{singular} && \text{smooth} && \text{orbi-singular} }


    • CommentRowNumber47.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 3rd 2018

    One would imagine that some branch of geometry had glimpsed this adjoint triple already, and so had some good names. Algebraic geometry seems particularly rich in terms for singularities, resolutions and quotients. (I see they’re not averse to value judgements. Along with GIT and geometric quotient, they also have ’good’ quotient, here.)

    Does tying names and symbols to a specific choice of SingularitiesSingularities raise an issue, if you ever want to generalise to other global families?

    Might be worth a question at MO. People get crotchety when others run counter to existing terminology, as we saw with ’parametrized category theory’.

    • CommentRowNumber48.
    • CommentAuthorUrs
    • CommentTimeDec 3rd 2018

    Here is a rough idea:

    For the geometric cohesion, shape and flat are something one is usually interested in actually applying to a given objects, while the role of sharp is more indirect, for instance we want to know is a an object is a subobject of its sharp aspect.

    Here for the singularities we have that flat (smooth homotopy quotient, “\subset”) and sharp (orbifold quotient, \prec) are the two which we want to apply directly, while now the shape (bad quotient, “<\lt”) is the outlier.

    Maybe this suggests that the interesting use of the bad quotient modality “<\lt” is in turn more indirect. Maybe we should ask if it means anything special if an orbifold 𝒳\mathcal{X} is such that 𝒳<(𝒳)\mathcal{X} \to \lt(\mathcal{X}) is nn-connected, or something like that.

    • CommentRowNumber49.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 3rd 2018

    So in the case of \sharp, the extensive/intensive quantity story is interesting, whether a type is anti-modal or there’s a mono into the sharp version.

    Since <\lt is left adjoint, you’re looking then to variations of 𝒳<(𝒳)\mathcal{X} \to \lt(\mathcal{X}) as epi?

    How about the anti-modal aspect? But maybe having a quantity reduce to *\ast in the codomain place, as with extensive quantities, is more important than a space reducing to *\ast under <\lt.

    • CommentRowNumber50.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 3rd 2018

    Is it possible that less structure will be seen in the case of singular modalities? I mean, shape, forming an \infty-groupoid, involves morphisms at all levels, whereas <\lt is just tidying away the action of a 1-group.

    I recall asking Charles Rezk about using higher groups rather than just 1-groups for global equivariance, and he thought it would work, so long as compactness is maintained. Using the (,1)(\infty, 1)-category of finite \infty-groups would make <\lt do a little more work.

    • CommentRowNumber51.
    • CommentAuthorUrs
    • CommentTimeDec 3rd 2018
    • (edited Dec 3rd 2018)

    This doesn’t seem the right perspective to me. Neither a definition nor a theorem necessarily becomes better just by making it more complicated. Beware of disregard for the trivial.

    • CommentRowNumber52.
    • CommentAuthorUrs
    • CommentTimeDec 3rd 2018
    • (edited Dec 3rd 2018)

    But to see where things should go we should come back to your question in #41:

    Could this duality of dualities partake in any larger diagram?

    If I write

    SuperSingularitiesSuperFormalCartesianSpaces×Groupoids cn,fin SuperSingularities \;\coloneqq\; SuperFormalCartesianSpaces \times Groupoids_{cn, fin}


    HSh (SuperSingularities) \mathbf{H} \coloneqq Sh_\infty(SuperSingularities)

    carries the familiar progression of modal operators

    id id | Rh \ & | ʃ / * \array{ id &\dashv& id \\ \vee &\vert& \vee \\ \rightrightarrows &\dashv& \rightsquigarrow &\dashv& Rh \\ && \vee &\backslash& \vee \\ && \Re &\dashv& \Im &\dashv& \& \\ && && \vee &\vert& \vee \\ && && &#643; &\dashv& \flat &\dashv& \sharp \\ && && && \vee &/& \vee \\ && && && \emptyset &\dashv& \ast }

    and carries in addition the modal operators

    id id | < / * \array{ id &\dashv& id \\ \vee &\vert& \vee \\ \lt &\dashv& \subset &\dashv& \prec \\ && \vee &/& \vee \\ && \emptyset &\dashv& \ast }

    and they commute with each other. But currently they don’t communicate with each other.

    Hm. What might be a good 2-category that Groupoids cn,finGroupoids_{cn,fin} is co-reflective in?

    • CommentRowNumber53.
    • CommentAuthorDavid_Corfield
    • CommentTimeDec 4th 2018

    Does the “global equivariant” world have anything to do with that of intersection cohomology, perverse sheaves, stratified spaces, etc.?

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