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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 $\mathcal{G}$ 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)
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 $R^n \sslash G_i$ ?
Does the cocycle on the orbifold restrict to a $G_i$-equivariant Bredon cocycle on the $i$th 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?
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.
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 $R^n \sslash G_i$ ?
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.
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.
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 $S^4$ with given $SU(2)$-action as an orbispace, in the above sense. b) Regard an orbifold $\mathcal{X}$ with isotropy groups being finite subgroups of $SU(2)$ 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:
$GlobalEquivariantCohomotopy(\mathcal{X}) \;\coloneqq\; \pi_0 Maps_{Orbispaces}( \mathcal{X}, S^4\sslash SU(2) ) \;\simeq\; \pi_0 Maps_{GlobalSpaces/\mathcal{N}}( \mathcal{X}, S^4\sslash SU(2) ) \,.$where the equivalence is by Charles’s result in Global Homotopy Theory and Cohesion (e.g. top of p. 4).
What is $\mathcal{N}$ here?
Where’s a good spot to note that $\mathcal{N}$ 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.
Ah, I see: $\mathcal{N}$ is the terminal orbispace.
Yes, $\mathcal{N}$ is the group object in $\infty$-presheaves over the full subcategory on $\mathbf{B}G$-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 $G$-equivariant for given $G$ if it lifts to a map in the slice over $\mathbf{B}G$ (by the discussion at infinity-action). To make this work for all $G$ 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 $\mathcal{N}$ 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 $SU(2)$, and $SU(2)$ is the universal among these, in the suitable generalized sense. Concretely this comes down to the elementary statement that if $\mathcal{X}$ is an orbifold with isotropy groups being finite subgroups of $\mathrm{SU}(2)$, then there is a canonical faithful morphism $\mathcal{X} \to \mathbf{B} SU(2)$, and for $A$ any space with an $SU(2)$-action, a global $SU(2)$-equivariant cocyle on $\mathcal{X}$ is simply a map of stacks $\mathcal{X} \longrightarrow A\sslash \mathbf{B}SU(2)$ in the slice over $\mathbf{B}SU(2)$.
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.
Does $\mathcal{F}$ have to satisfy any particular properties? I recall trying to see if the finite subgroups of $SU(2)$ 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.
Hi David R., thanks for highlighting this technical issue again, but here I just meant to illustrate the role of $\mathcal{N}$. By the nature of $\mathcal{N}$ we may slice over $\mathcal{N}$ instead of over $\mathbf{B}SU(2)$, 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.
Ok, good point.
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 $\mathcal{X}$ whose isotropy groups are any finite subgroup $G \overset{\iota}{\hookrightarrow} G_{glob}$ of a fixed compact Lie group $G_{glob}$. Such orbifolds carry canonical morphisms to the delooping stack $\mathbf{B}G_{glob}$, which are locally, for global quotients $\mathcal{X} \simeq X \sslash G$, given by
$\rho \;\colon\; X \sslash G \overset{ (X \to \ast) \sslash G }{\longrightarrow} \mathbf{B}G \overset{ \mathbf{B} \iota }{\longrightarrow} \mathbf{B}G_{glob} \,.$By the general discussion at ∞-action any such morphism exhibits an ∞-action of $G_{glob}$ on the homotopy fiber $hofib(\rho)$ of $\rho$, together with an equivalence
$(1) \phantom{AA} hofib(\rho)\sslash G_{glob} \;\simeq\; X \sslash G \,.$But that homotopy fiber is directly computed to be
$\begin{aligned} hofib(\rho) & \simeq \ast \times_{\mathbf{B}G_{glob}} \big( \mathbf{B}G_{glob} \big)^{\Delta[1]} \times_{\mathbf{B}G_{glob}} \mathbf{B}G \\ & \simeq \big( X \times G_{glob} \big) / G \\ & \simeq X \times_G G_{glob} \,, \end{aligned}$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 $( X \times_G K ) \sslash K \simeq X \sslash G$ for $G \hookrightarrow K \coloneqq G_{glob}$.
In conclusion, if $\mathcal{X}$ is any orbifold, i.e. not necessarily a global homotopy quotient $X_i \sslash G_i$, but locally of this form for each $G_i$ some finite subgroups of $G_{glob}$, then it comes with a canonical morphism of topological stacks $\mathcal{X} \overset{\rho}{\to} \mathbf{B}G_{glob}$, and so for $\mathcal{A} \overset{}{\to} \mathbf{B}G_{glob}$ any coefficient ∞-stack in the slice over $\mathbf{B}G_{glob}$, we may take the global equivariant orbifold cohomology to be given by homotopy classes of morphisms in the slice:
$H_{glob}(\mathcal{X}, \mathcal{A} ) \;\simeq\; \pi_0 \mathbf{H}_{/ \mathbf{B}G_{glob}} \big( \mathcal{X}, \mathcal{A} \big) \,.$Cocycles in this “global equivariant” cohomology are then such that on each chart of the form $U_i \sslash G_i$ they restrict to cocycles in $G_i$-equivariant cohomology of $U_i$, in a way that is compatible with the above re-identifications (eq:EquivalenceOfOrbifolds).
added further paragraphs to make the connection to the “Gepner-Henriques global Elmendorf theorem”:
Notice that if also the coefficient $\mathcal{A} \overset{}{\to} \mathbf{B} G_{glob}$ is faithful (0-truncated) as an object in the slice, then, by the orthogonality of the (n-connected, n-truncated) factorization system for $n = 0$, there is a contractible space of homotopies $\phi$ in the data for a cocycle
$\array{ \mathcal{X} && \overset{ \phantom{AA} c \phantom{AA} }{\longrightarrow} && \mathcal{A} \\ & {}_{\mathllap{\rho}}\searrow &\swArrow_{\phi}& \swarrow_{\mathrlap{f}} \\ && \mathbf{B}G_{glob} } \phantom{AAA} = \phantom{AAA} \array{ \mathcal{X} &\overset{ \phantom{A} c \phantom{A} }{\longrightarrow}& \mathcal{A} \\ {}^{=}\big\downarrow & \swArrow_{\exists !} & \big\downarrow{}^{\mathrlap{f}} \\ \mathcal{X} &\underset{ \phantom{A} \rho \phantom{A} }{\longrightarrow}& \mathbf{B}G_{glob} }$Moreover, in this case the cocyle morphism $c$ 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:
$\mathbf{H}_{/\mathbf{B}G_{glob}} \big( \mathcal{X}, \mathcal{A} \big) \;\simeq\; \mathbf{H}^{faith} \big( \mathcal{X}, \mathcal{A} \big) \,,$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 $Orb_G$ used in $G$-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 whose morphisms are the faithful/0-truncated morphisms between these (Henriques-Gepner 07, 4.1, Rezk 14, 4.5). Then any stack $\mathcal{X}$ (orbifold, orbispace) becomes an (∞,1)-presheaf $y \mathcal{X}$ over $Orb_{glb}$ by the evident “external Yoneda embedding”
$y \mathcal{X} \;\coloneqq\; \mathbf{H}^{faith}( \mathbf{B}G, \mathcal{X} ) \,.$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
$\mathbf{H}^{faith} \big( \mathcal{X}, \mathcal{A} \big) \;\simeq\; PSh_{\infty}(Orb_{glb}) ( y \mathcal{X}, y \mathcal{X} ) \,.$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
$H(\mathcal{X}, \mathcal{A}) \;\simeq\; \pi_0 PSh_\infty(Orb_{glb})( y\mathcal{X}, y \mathcal{a} ) \,.$dropped a comment also in the comment section here alongside Felix’s exposition “Differential Geometry in Modal HoTT”
I will rewrite that discussion again.
The point is really that we may regard any smooth $\infty$-groupoid $\mathcal{X} \in \mathbf{H} \coloneqq Sh_\infty(CartSp)$ as a presheaf $eq(\mathcal{X})$ of $GlobalOrbits = \{\text{finite groupoids} B K \text{with faithful functors}\}$ by
$B K \mapsto [Disc(B K), \mathcal{X}]_{\mathbf{H}}$and if $\mathcal{X}$ happens to be an orbifold, then this is its correct incarnation as a global equivariant thing. In the sense that for $G$ any finite group, $A \in G Space$ any $G$-space, regarded as
$Disc \delta_G A \;\in\; PSh_\infty( GlobalOrbits, \mathbf{H} )$the correct orbifold cohomology is
$PSh_\infty( GlobalOrbits, \mathbf{H} )( e(\mathcal{X}), Disc \delta_G A ) \,.$and this expression will automatically first regard $\mathcal{X}$ as a $G$-space, if possible.
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 $\mathbf{H}$ be a differentially cohesive $\infty$-topos. Write
$Singularities \;\coloneqq\; Groupoids_{1,fin}^{cn}$for the $(2,1)$-category of finite connected groupoids and, following Charles Rezk, write
$\mathbb{B}G \;\in\; Singularities$for the object given by the delooping groupoid on some finite group $G$.
The notation is to distinguish from the delooping groupoid $\mathbf{B}G \in \mathbf{H}$, 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 $\mathbf{B}G =\ast \sslash G \in \mathbf{H}$, 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 $\ast / G = \ast$, 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 $\mathbf{B} \mathbb{Z}_2 = \mathbb{R}P^\infty$ 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:
Write
$\mathbf{H}_{sing} \;\coloneqq\; Sh_\infty( Singularities, \mathbf{H} )$Remark on terminology. The category $Singularities$ is usually called “$Orb$ version 1” (Gepner-Henriques) or “$GLob$ (Rezk)” or just “$Orb$” (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 $(2,1)$-category are the local models for orbifold singularities. In particular if $SynthCartSpace$ is a site of definition of $\mathbf{H}$ with objects such as $\mathbb{R}^n$, then a site of definition of $\mathbf{H}_{sing}$ is $Singularities \times SynthCartSpace$ with objects such as $\mathbb{R}^n \times \mathbb{B}G$, which are manifestly the local model spaces for orbifold singularities.
Since $Singularities$ is a small $\infty$-site with finite products, we have immediately that $\mathbf{H}_{sing}$ is cohesive over $\mathbf{H}$, as in section 5.1 of Charles Rezk’s Global Homotopy Theory and Cohesion. I’ll denote the corresponding modalities with subscript ${}_{sing}$.
Now:
For $V \in \mathbf{H} \overset{Disc_{sing}}{\hookrightarrow} \mathbf{H}_{sing}$ a group object, we may consider V-manifolds $X \in \mathbf{H} \overset{Disc_{sing}}{\to} \mathbf{H}_{sing}$.
For $G$ a finite group, say that such a $V$-manifold is an orbifold with singularities/isotropy groups in $G$ if there is a 0-truncated (i.e. “faithful”) morphism
$(X \longrightarrow \mathbf{B}G) \;\in\; \mathbf{H} \overset{Disc_{sing}}{\hookrightarrow} \mathbf{H}_{sing}$Picking any such is a real choice, I suppose. But let’s pick one and regard the result as
$\mathcal{X} \;\in\; \mathbf{H}_{/\mathbf{B}G} \overset{Disc_{sing}}{\hookrightarrow} \left( \mathbf{H}_{sing}\right)_{/\mathbf{B}G}$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 $\mathbf{B}G$ 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:
$\begin{aligned} \sharp_{sing} \mathcal{X} & = coDisc_{sing} \Gamma_{sing} Disc_{sing} (X \to \mathbf{B}G) \\ & \simeq coDisc_{sing} (X \to \mathbf{B}G) \\ & =coDisc_{sing}(X) \to \mathbb{B}G \end{aligned}$To see what the last line is, use that
$\Gamma_{sing}(\mathbb{B}G) \;=\; Singularities( \mathbb{B}1 , \mathbb{B}G) \;=\; \mathbf{B}G$and the adjunction $(\Gamma_{sing} \dashv coDisc_{sing})$:
$coDisc_{sing}(X) \;\colon\; \mathbb{R}^n \times \mathbb{B}G \;\mapsto\; \mathbf{H}( \mathbb{R}^n, \times \mathbf{B}G )$In words: $coDisc_{sing}$ reads in an orbifold regarded as a smooth groupoid, and then replaces all its $\mathbf{B}G$-singularities with the refined $\mathbb{B}G$-singularties.
In particular $coDisc_{sing}(\mathbf{B}G) \simeq \mathbb{B}G$.
Similarly it is immediate that with $X \to \mathbf{B}G$ being faithful (0-truncated), so is $coDisc_{sing}(X) \to \mathbb{B}G$.
But this means, by section 4 in Global Homotopy Theory and Cohesion, that our improved orbifold $\sharp_{sing}(\mathcal{X})$ with singularities in $G$ is indeed an object now in geometric $G$-equivariant homotopy theory (“geometric” because we are over the differentially cohesive base $\infty$-topos $\mathbf{H}$ instead of just over $Groupoids_\infty$.)
In particular now for
$\mathcal{A} \;in\; \big( (\mathbf{H}_{sing})_{/\mathbb{B}G} \big)_{faith}$any coefficient object, the correct Bredon-equivariant geometric/differential cohomology of $\mathcal{X}$ is
$H_{G}(\mathcal{X}, \mathcal{A}) \;\coloneqq\; \pi_0 \mathbf{H}_{sing}\left( \sharp_{sing} \mathcal{X}, \mathcal{A}\right) \,.$Voilà.
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.
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?
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 (“$Glob$” $\rightsquigarrow$ “$Singularities$”) 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 $\mathbb{B}G$.
Removing the magnifying glass, what one sees with the bare eye depends on how one squints:
The physicist says that what he sees is a singular point, but a point after all. That’s $\ast = \ast / G$.
The Lie geometer says that what he sees is a point with a trivial $G$ action, a groupoid $\mathbf{B}G =\ast \sslash G$.
These two aspects are two opposite extreme aspects of the orbifold singularity $\mathbb{B}G$, 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:
$\array{ && { \text{orbifold singularity} \atop {\mathbb{B}G} } \\ & {}^{\mathllap{ʃ_{sing}}}\swarrow & {{\phantom{A}} \atop { \text{opposite extreme} \atop \text{aspects of orbifold singularity} }} & \searrow^{\mathrlap{ \flat_{sing} }} \\ { \text{plain quotient} \atop {\ast = \ast/G} } && && { \text{homotopy quotient} \atop { \mathbf{B}G = \ast \sslash G } } }$So we have points-to-pieces, $\flat_{sing} \mathcal{X} \to \mathcal{X} \to ʃ_{sing} \mathcal{X}$. 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 $Singularities$ could be any global family.
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 $n$-connected for various $n$.
So it is always $(-1)$-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 $Singularities$ 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 $Singularities$ should hence be finite connected groupoids and nothing else. Also the nice story in #19 with refined orbifolds being $coDisc_{sing}$ of their Lie groupoid incarnation only works for discrete groups.
Another natural thing to wonder about is the differential cohomological diagram in this case. What kind of fracturing goes on for stable singular spaces?
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.
Very nice! (Unfortunately I have no comment at present of of more substance)
“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 $Glo$ there is an enriched category whose objects are compact Lie groups, so include a word on your choice of $Singularities$.
Thanks!!
Will anything interesting happen as you refine the $\mathbf{H}$ to a super differential cohesive version? There must be ’super-orbifolds’.
Some google hits for that, but not as many as might be expected.
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.
Look forward to seeing it all written out.
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 $ʃ_{sing} \dashv \flat_{sing} \dashv \sharp_{sing}$, but there must be better choices.
It might be good if the symbols used could suggest the idea that $ʃ_{sing}$ produces something that is “too small” or “too rigid”, while $\flat_{sing}$ produces something that is “too big” or “too fuzzy”, compared to the real thing.
I guess one advantage of the current way is that $\mathbf{H}_{sing}$ itself is cohesive over $\infty Grpd$ if $\mathbf{H}$ is, so that a composition such as $\Gamma \cdot \Gamma_{sing}$ is just another form of $\Gamma$.
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.
so that a composition such as $\Gamma \cdot \Gamma_{sing}$ is just another form of $\Gamma$.
Ah, I am not looking for alternatives to the symbols $\Pi \dashv Disc \dashv \Gamma \dashv coDisc$, but for alternative to the symbols for the corresponding modal operators $ʃ \dashv \flat \dashv \sharp$
Maybe the symbol
$\succ$could be used instead of $\sharp_{sing}$, suggesting a singular tip (in LaTeX I wold maybe rotate it 90 degrees), while
$\smile$could then be used for $\flat_{sing}$, suggesting smoothness in the sense of absence of singularities
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.
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.
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?
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.
“Naïve quotient”?
“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 $C^k$-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
$\subset & \array{ \lt &\dashv& \subset &\dashv& \prec \\ \text{non-smooth} && \text{smooth} && \text{singular} }$or
$\array{ \lt &\dashv& \subset &\dashv& \prec \\ \text{singular} && \text{smooth} && \text{cosingular} }$or something like this.
How about “orbits”? Or are you looking for an adjective?
But “orbits” means something else, I’d think. Maybe “orbifold-singular”?
$\array{ \lt &\dashv& \subset &\dashv& \prec \\ \text{singular} && \text{smooth} && \text{orbi-singular} }$?
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 $Singularities$ 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’.
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 $\mathcal{X}$ is such that $\mathcal{X} \to \lt(\mathcal{X})$ is $n$-connected, or something like that.
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 $\mathcal{X} \to \lt(\mathcal{X})$ 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 $\lt$.
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 $(\infty, 1)$-category of finite $\infty$-groups would make $\lt$ do a little more work.
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.
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
$SuperSingularities \;\coloneqq\; SuperFormalCartesianSpaces \times Groupoids_{cn, fin}$then
$\mathbf{H} \coloneqq Sh_\infty(SuperSingularities)$carries the familiar progression of modal operators
$\array{ id &\dashv& id \\ \vee &\vert& \vee \\ \rightrightarrows &\dashv& \rightsquigarrow &\dashv& Rh \\ && \vee &\backslash& \vee \\ && \Re &\dashv& \Im &\dashv& \& \\ && && \vee &\vert& \vee \\ && && ʃ &\dashv& \flat &\dashv& \sharp \\ && && && \vee &/& \vee \\ && && && \emptyset &\dashv& \ast }$and carries in addition the modal operators
$\array{ id &\dashv& id \\ \vee &\vert& \vee \\ \lt &\dashv& \subset &\dashv& \prec \\ && \vee &/& \vee \\ && \emptyset &\dashv& \ast }$and they commute with each other. But currently they don’t communicate with each other.
Hm. What might be a good 2-category that $Groupoids_{cn,fin}$ is co-reflective in?
Does the “global equivariant” world have anything to do with that of intersection cohomology, perverse sheaves, stratified spaces, etc.?
Any further thoughts on #52? It would be good to have a synthetic description of the equivariant modalities for your meeting talk.
Any further thoughts on #52?
Dan tells me he checked that for fixed finite group there is a reasonable Grothendieck topology on the category of smooth manifolds with $H$-action for $H$ a subgroup of $G$, such that there is a cover-preserving fully faithful functor $SmthMfd \hookrightarrow G SmthMfd$ which should have a right adjoint given by forming $G$-fixed points.
Something like this could generalize the plain product site $SmthMfd \times GlobalOrbitCat$ in a way that would make the two ingredients not just run in parallel, and it should then make the corresponding adjoint modalities organize in a connected progression.
But to explore this further I’d need some indication for a problem that this solves, which isn’t solved by using the product site structure.
Thanks for the update!
added publication data for
We now have a first version of a note developing this idea of proper orbifold cohomology in singular-cohesive $\infty$-toposes:
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H. Sati, U. Schreiber:
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Abstract. The concept of orbifolds should unify differential geometry with equivariant homotopy theory, so that orbifold cohomology should unify differential cohomology with proper equivariant cohomology theory. Despite the prominent role that orbifolds have come to play in mathematics and mathematical physics, especially in string theory, the formulation of a general theory of orbifolds reflecting this unification has remained an open problem. Here we present a natural theory argued to achieve this. We give both a general abstract axiomatization in higher topos theory, as well as concrete models for ordinary as well as for super-geometric and for higher-geometric orbifolds. Our first main result is a fully faithful embedding of the $2$-category of orbifolds into a singular-cohesive $\infty$-topos whose intrinsic cohomology theory is proper globally equivariant differential generalized cohomology, subsuming traditional orbifold cohomology, Chen-Ruan cohomology, and orbifold K-theory. Our second main result is a general construction of orbifold étale cohomology which we show to naturally unify (i) tangentially twisted cohomology of smooth but curved spaces with (ii) RO-graded proper equivariant cohomology of flat but singular spaces. As a fundamental example we present J-twisted orbifold Cohomotopy theories with coefficients in shapes of generalized Tate spheres. According to “Hypothesis H” this includes the proper orbifold cohomology theory that controls non-perturbative string theory.
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Comments are welcome. Please grab the latest version of the file from behind the above link.
The bibliographic reference [Pav19] conflates 3 papers into 1:
A) In Definition 2.19 it should refer to my forthcoming paper “A cartesian combinatorial model structure on diffeological spaces and smooth sets” (I am currently finalizing the proof of the existence of a model structure, the proof that all manifolds are cofibrant and the model structure is cartesian will be written down next.)
B) In Example 3.18(iii) it can refer to my paper https://arxiv.org/abs/1912.10544, which is joint work with two other authors.
C) There is also my own manuscript “Structured Brown representability via concordance”, which proves a version for sheaves valued in general ∞-categories, as opposed to just spaces, and has a much easier proof than B). I am currently waiting for B) to be accepted in a journal, to ensure that C) does not interfere with the publication of B).
Just dipping in
In (153), $W$ has become $V$.
In Lemma 3.23(i), $k$ takes the value $0$, when in Defn. 3.22 it specified $k \geq 1$.
In Lemma 3.23(ii), what’s the domain of $Rdcd_{\infty}$?
$LcllCnstnt_X$ in (164) becomes $LcllConst_X$
Around (180): degenracy
an examples
Defn 3.40 (ii), where’s the middle $id$ going in $f_i \times id \times id$?
Hi Dmitri,
thanks for taking a look!
re B) Sorry for the glitch of omitting your coauthors on arXiv:1912.10544. Fixed now. (This was one of the first references added, when the document was all rough, and then apparently this item was missed when polishing things up.)
re C) Okay, I have added pointer to you solo note, too. Should there be a date to go with this?
In the latest version now the joint article appears as [BEBP19] and your solo version as [Pav].
re A) indeed, I was going to ask you about that, following up on the long discussion we had here.
First to say that our Def. 2.19 just recalls the definition of extended simplices.
The real deal is in Example 5.7 (i) where we claim, currently without proof, that the cohesive shape of a Fréchet manifold is equivalently its ordinary topological shape.
Here are two options:
a) If there is no certainty of proof available for this at the time that we post to the arXiv, then we will just remove the adjective “Fréchet” and have the statement for finite dim manifolds. Nothing in our article depends on the Fréchet-generality, it would just be neat to mention, if true.
b) If you do confidently claim that you have proof for this, writeup upcoming, then we’ll add a Lemma saying this, attributed to your upcoming note, and then make Example 5.7 (i) point to that Lemma.
So how do I read your remark A) ? Are you confident that for $X \in FrechetManifolds$ we have that its cohesive shape is equivalently its ordinary topological shape? Please do let me know.
Hi David,
thanks for catching these. All fixed now.
Except for the domain of $Rdcd_\infty$. Yeah, this is notationally a bit of a cheat. This is meant to be the colimiting coprojection of that sequence. So the domain would be any one of the objects at some finite stage. But making that notionally more correct would give a typographical mess. So I thought I could hide that in the innocent-looking ellipses. ;-)
Some more:
orbi-sing ular, in the table on p.7.
There’s an adjunction symbol in the diagram (23) with no left adjoint mentioned.
Remark 3.63: dicrete
Why did you revert to Lawvere’s ’chaotic’ rather than ’codiscrete’?
Thanks again for catching typos. Fixed now.
It seems that a positive adjective like “chaotic” works better with those abbreviated terms than a negative one like “co-discrete” (which few topologists would ever say?!) or “in-discrete”.
It seems that Lawvere really did want to think about chaos in those terms, but I can’t see that anybody made anything of it.
Another typo:
In Proposition 3.49, in the diagram you have the $\mathbf{H}$ and $\mathbf{B}$ with the smooth subscripts switched.
But “chaotic topology” is a fairly standard term, no?
I hadn’t realised it went further back. SGA4-1, 1.1.4., it seems.
Re #61:
re C) Okay, I have added pointer to you solo note, too. Should there be a date to go with this?
I think you can just refer to it as “Manuscript in preparation, 2020.”
Here are two options: a) If there is no certainty of proof available for this at the time that we post to the arXiv, then we will just remove the adjective “Fréchet” and have the statement for finite dim manifolds. Nothing in our article depends on the Fréchet-generality, it would just be neat to mention, if true. b) If you do confidently claim that you have proof for this, writeup upcoming, then we’ll add a Lemma saying this, attributed to your upcoming note, and then make Example 5.7 (i) point to that Lemma. So how do I read your remark A) ? Are you confident that for X∈FrechetManifoldsX \in FrechetManifolds we have that its cohesive shape is equivalently its ordinary topological shape? Please do let me know.
I do intend to have it ready before the end of the summer, unless other commitments intervene. But I also want to add a proof that the model structure is cartesian and all manifolds are cofibrant, and this may take a bit more time, possibly extending in the fall.
I would suggest a fusion of a) and b): you could claim it first for finite-dimensional spaces, and then add a separate remark for Frechet manifolds. (Something like: “In a forthcoming paper […], Theorem 1.1 implies a generalization of the above result to Frechet manifolds.” The existence of model structures will definitely be “Theorem 1.1” in my paper.
David,
thanks for catching that typo in the first diagram in Prop. 3.49, that would have been an unfortunate typo to keep around. Fixed now.
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Dmitri,
okay, sure.
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I should say that, in any case, I am leaving for a 10 day vacation with family, tomorrow morning, which means that reactions from my side to further comments, will be slow to non-existent in the next days.
Have a great time!
Something I’d like to see to help understand these singular modalities is something like the discussion of the two unity transformations for discreteness/continuity and repulsion/cohesion at adjoint modality. So how to think of a space located between the smooth and the singular and then between the smooth and the orbi-singular.
In Example 4.20, presumably that’s
any $V \in Groups(\mathbf{H})$
Typos:
subategory; singlarity; the the (twice); Gropoids; $Snglr \dashv NnOrbSnglrs$.
Thanks! Fixed now.
By the way, we are doing a semester-long seminar on this: https://dmitripavlov.org/homotopy
That’s nice!
There is something weird about how Proposition 2.34 in Proper Orbifold Cohomology is formulated. It talks about the “terminal inverse image functor”.
But inverse image functors go in the opposite direction from geometric morphisms. In particular, the terminal ∞-topos in the category of toposes and geometric morphisms because the initial ∞-topos in the category of toposes and inverse image morphisms. Equivalently, inverse image functors T→H, where H is a fixed ∞-topos and T is an arbitrary ∞-topos, form a slice ∞-category over H, and ∞Grpd→H is the initial object in this ∞-category.
Thanks for highlighting. Sure, I have changed to “inverse base geometric morphism” (in the pdf file here), to match the terminology actually introduced in 2.43. (Hm, and maybe ordering of these Propositions should be changed. But I left that as is for the moment.)
In Proposition 2.48, it appears that $B$ and $X$ are used interchangeably for the same entity.
Thanks for catching! I have fixed it (in the pdf copy here).
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