nLab
orbifold cohomology

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Higher geometry

Contents

Idea

Since the crucial extra structures carried by an orbifold are

  1. geometric structure (e.g. topological, algebro-geometric, differential-geometric, super-geometric, etc.)

  2. singularities

the cohomology of orbifolds should be such as to provide invariants which are sensitive not just to the underlying plain homotopy type of an orbifold (its shape) but also to this extra structure. This means that orbifold cohomology should, respectively, unify

  1. geometric cohomology (e.g. sheaf hypercohomology, differential cohomology, etc.)

  2. equivariant cohomology in its fine form of Bredon cohomology

in the sense that geometric cohomology is recovered away from the orbifold singularities and equivariant cohomology is recovered right at the singularities, while globally orbifold cohomology provides a unification of these two aspects.

Since any concept of cohomology (as discussed there) is effectively equivalent to the choice of ambient (∞,1)-topos, the question of defining orbifold cohomology is closely related to the question of how exactly to define the (∞,1)-category of orbifolds (usually a (2,1)-category) in the first place. This question is notoriously more subtle than the simple intuitive idea of orbifolds might suggest, as witnessed by the convoluted history of the concept (see e.g. Lerman 08, Introduction).

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Deficiency of orbifolds as geometric groupoids

A proposal popular among Lie theorists (Moerdijk-Pronk 97) is to regard an orbifold with local charts U iG iActionsU_i \in G_i Actions (actions of some group on some local model space) as the geometric stack obtained by gluing the corresponding homotopy quotients/quotient stacks U iG iU_i \!\sslash \!G_i. If H\mathbf{H} is the ambient cohesive (∞,1)-topos in which this takes place (for instance H=\mathbf{H} = Smooth∞Groupoids, SuperFormalSmooth∞Groupoids, etc.) and if GGrpDiscGrp(H)G \in Grp \overset{Disc}{\hookrightarrow} Grp(\mathbf{H}) is a discrete group in which all the isotropy groups of the orbifold are contained, this gives an object

(𝒳 faith BG faith)(H /BG) 0H /BG \left( \array{ \mathcal{X} \\ \downarrow^{\mathrlap{faith}} \\ \mathbf{B}G } \phantom{{}^{faith}} \right) \;\in\; \left( \mathbf{H}_{/\mathbf{B}G} \right)_{\leq 0} \hookrightarrow \mathbf{H}_{/\mathbf{B}G}

in the slice (∞,1)-topos over the delooping BG=*G\mathbf{B}G = \ast \sslash G of GG, which is still a 0-truncated object, reflecting that as a functor of groupoids the morphism 𝒳BG\mathcal{X} \to \mathbf{B}G is a faithful functor.

Accordingly, if this is – or were – the correct formalization of the nature of orbifolds 𝒳\mathcal{X}, then the corresponding orbifold cohomology has coefficients given by objects 𝒜H /BG\mathcal{A} \in \mathbf{H}_{/\mathbf{B}G} and cohomology sets being the connected components of the (∞,1)-categorical hom-spaces

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

This concept of orbifold cohomology does fully reflect the geometric nature of orbifolds. It also reflects some equivariance aspect. For example if 𝒳=*G\mathcal{X} = \ast \sslash G is the one-point orbifold with singularity given by a finite group GG, and if VGRepresentationsV \in G Representations is a linear representation, with K(V,n)GGroupoids /BGDiscH /BGK(V,n)\sslash G \in \infty Groupoids_{/\mathbf{B}G} \overset{Disc}{\hookrightarrow} \mathbf{H}_{/\mathbf{B}G} its Eilenberg-MacLane space, then

H BG(BG,K(V,n)G)H grp n(G,V) H_{\mathbf{B}G}\big( \mathbf{B}G, K(V,n)\sslash G \big) \;\simeq\; H^n_{grp}(G,V)

is the group cohomology of GG with coefficients in VV.

However, this definition does not reflect Bredon-equivariant cohomology around the orbifold singularities. Instead, it really given (geometric/stacky refinement) of cohomology with local coefficients.

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Lift of orbifolds to geometric global homotopy theory

Hence the proposal of Moerdijk-Pronk 97, that an orbifold should be regarded as a certain geometric stack, is missing something. It was briefly suggested in Schwede 17, Introduction, Schwede 18, p. ix-x that the missing aspect is provided by global equivariant homotopy theory, but details seem to have been left open.

Here we discuss how to define the required orbifold cohomology in detail and in general. We combine the differential cohesion for the geometric aspect with the cohesion of global equivariant homotopy theory that was observed and highlighted in Rezk 14.

The following may serve as intuition for the issue with the nature of orbifolds:

Envision the picture of an orbifold singularity and hold a mathemagical magnifying glass over the singular point. Under this magnification you can see resolved the singular point as a fuzzy fattened point, to be called 𝔹G\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. This is the plain quotient *=*/G\ast = \ast / G.

  • The Lie geometer says that what he sees is a point transforming under the GG-action that fixes it, hence the homotopy quotient groupoid BG=*G\mathbf{B}G =\ast \sslash G.

These are two opposite extreme aspects of the orbifold singularity 𝔹G\mathbb{B}G, but the orbifold singularity itself is more than both of these aspects. The real nature of an orbifold singularity is in fact a point, not a big classifying space BG\mathbf{B} G (recall that already B 2=P \mathbf{B}\mathbb{Z}_2 = \mathbb{R}P^\infty), but it is a point that also remembers the group action, for that characterizes how the singularity is being singular.

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 } } }

Definition

The definition of orbifold cohomology below in Def. is the canonical cohomology in a slice of the globally equivariant homotopy theory H sing\mathbf{H}_{sing} of the given ambient cohesive (∞,1)-topos H\mathbf{H}.

For completeness, we first introduce/recall H sing\mathbf{H}_{sing} in Prop. below, as well as its slices to GG-equivariant homotopy theory H G\mathbf{H}_G (Def. below), following Rezk 14.

The key point in the following is the identification of orbifolds as the 0-truncatedSingularitiesSingularities-codiscrete objects” in H sing\mathbf{H}_{sing} (Def. below), which is consistent in that under passage to shapes, i.e. the underlying bare globally equivariant homotopy types, it reproduces the standard embedding of G-spaces into globally equivariant homotopy theory (Example below.)

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Globally equivariant cohesive toposes

We consider here the evident refinement to geometric cohomology hence to differential cohomology (cohesive ∞-stacky sheaf cohomology) of globally equivariant homotopy theory, in Prop. below.

In order to bring out the conceptual appearance of orbifolds further below, we take the liberty of referring to what otherwise is known as the “global orbit category” instead as the categories of singularities (Def. below).

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Definition

(category of singularities)

Write

SingularitiesGroupoids 1,fin cnGroupoids Singularities \;\coloneqq\; Groupoids^{cn}_{1,fin} \hookrightarrow Groupoids_\infty

for the (2,1)-category of connected finite groupoids. A skeleton has objects labeled by finite groups GG, and we will denote these objects

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

to distinguish them from their image as delooping groupoids BG\mathbf{B} G \in ∞Grpd. (As we consider (∞,1)-presheaves on SingularitiesSingularities with values in ∞Groupoids, in Prop. below, these two objects become crucially different, albeit closely related.)

Remark

The category SingularitiesSingularities in Def. , when generalized from finite groups to compact Lie groups, is called

Proposition

(global equivariant homotopy theory cohesive over base (∞,1)-topos)

Let H\mathbf{H} be any (∞,1)-topos and consider the (∞,1)-category of (∞,1)-presheaves on the category of singularities (Def. ) over the base (∞,1)-topos H\mathbf{H}, hence the (∞,1)-functor (∞,1)-category

H singSh (Singularities,H)=Funct(Singularities op,H). \mathbf{H}_{sing} \;\coloneqq\; Sh_\infty\big( Singularities, \mathbf{H}\big) \;=\; Funct\big( \Singularities^{op}, \mathbf{H}\big) \,.

This is a cohesive (∞,1)-topos over the base (∞,1)-topos H\mathbf{H} in that the global section-geometric morphisms enhances to an adjoint quadruple of adjoint (∞,1)-functors

(1)(Π singDisc singΓ singcoDisc sing):H singH \big( \Pi_{sing} \dashv Disc_{sing} \dashv \Gamma_{sing} \dashv coDisc_{sing} \big) \;\colon\; \mathbf{H}_{sing} \leftrightarrow \mathbf{H}

such that

  1. Disc sing,coDisc sing:HH singDisc_{sing}, coDisc_{sing} \;\colon\; \mathbf{H} \to \mathbf{H}_{sing} are fully faithful (∞,1)-functors;

  2. Π sing\Pi_{sing} preserves finite products.

hence inducing an adjoint triple of adjoint modalities

ʃ sing sing sing:H singH sing ʃ_{sing} \dashv \flat_{sing} \dashv \sharp_{sing} \;\colon\; \mathbf{H}_{sing} \to \mathbf{H}_{sing}

(“shape”, “flat”, “sharp” for singularities).

Moreover, for GG a finite group regarded under the inclusion

GGrpDiscGrp(H)Disc singGrp(H sing) G \in Grp \overset{Disc}\hookrightarrow Grp(\mathbf{H}) \overset{Disc_{sing}}{\hookrightarrow} Grp\left(\mathbf{H}_{sing}\right)

and writing

BGH sing \mathbf{B}G \in \mathbf{H}_{sing}

for its delooping under Grp(H sing)BH cn */Grp\left( \mathbf{H}_{sing} \right) \underoverset{\simeq}{\mathbf{B}}{\longrightarrow} \mathbf{H}^{\ast/}_{cn},

in constrast to the (∞,1)-Yoneda embedding

𝔹GySh (Singularities,Grpd)DiscSh (Singularities,H) \mathbb{B}G \overset{y}{\longrightarrow} Sh_\infty\left( Singularities, \infty \mathrm{Grpd}\right) \overset{Disc}{\longrightarrow} Sh_\infty\left( Singularities, \mathbf{H}\right)

we have

(2)ʃ sing𝔹G */G=* sing𝔹G *G=BG \begin{aligned} ʃ_{sing} \mathbb{B}G & \simeq\; \ast/G = \ast \\ \flat_{sing} \mathbb{B}G &\simeq\; \ast \sslash G = \mathbf{B}G \end{aligned}
Proof

This is immediate by general properties of left/right (∞,1)-Kan extension, using the evident fact that SingularitiesSingularities has finite products (the terminal object is 𝔹1\mathbb{B}1 and the binary Cartesian product is give by forming direct product groups: (𝔹G 1)×(𝔹G 2)𝔹(G 1×G 2)\left(\mathbb{B}G_1\right) \times \left( \mathbb{B}G_2\right) \simeq \mathbb{B}\left( G_1 \times G_2\right) ). The directly analogous 1-categorical argument is at infinity-cohesive site.

Example

For H=\mathbf{H} = ∞Groupoids the cohesion of Prop. is that of plain globally equivariant homotopy theory (Rezk 14, 4.1), i.e. without any geometric determination (geometrically discrete ∞-groupoids).

Conversely we have:

Proposition

(orbifold singularities are the codiscrete aspect of homotopy quotients)

Let H\mathbf{H} itself be a cohesive (∞,1)-topos over ∞Groupoids

(ΠDiscΓcoDisc):HGroupoids. (\Pi \dashv Disc \dashv \Gamma \dashv coDisc) \;\colon\; \mathbf{H} \leftrightarrow \infty Groupoids \,.

Then in the situation of Prop.

coDisc singBG𝔹G. coDisc_{sing} \mathbf{B}G \;\simeq\; \mathbb{B}G \,.
Proof

Let 𝒮\mathcal{S} be a cohesive (∞,1)-site of definition for H\mathbf{H}, so that

H singSh (𝒮×Singularities,Groupoids) \mathbf{H}_{sing} \;\simeq\; Sh_\infty \left( \mathcal{S} \times Singularities \;,\; \infty Groupoids \right)

and Π(S)*\Pi(S) \simeq \ast for S𝒮yHS \in \mathcal{S} \overset{y}{\hookrightarrow} \mathbf{H}.

Then as (∞,1)-presheaves regarded this way we have

coDisc singBG:S×𝔹K H sing(S×𝔹K,coDisc singDiscBG) H(S×Γ sing(𝔹K)BK,DiscBG) Groupoids(Π(S×BK)BK,BG) Groupoids(BK,BG) H sing(𝔹K,𝔹G) H sing(S×𝔹K,𝔹G) \begin{aligned} coDisc_{sing} \mathbf{B}G \;\colon\; S \times \mathbb{B}K & \mapsto \mathbf{H}_{sing}\left( S \times \mathbb{B}K, coDisc_{sing} Disc \mathbf{B}G \right) \\ & \simeq \mathbf{H}\big( S \times \underset{ \simeq \mathbf{B}K }{ \underbrace{ \Gamma_{sing}\left( \mathbb{B}K\right) }} , Disc \mathbf{B}G \big) \\ & \simeq \infty Groupoids\big( \underset{ \simeq \mathbf{B}K }{ \underbrace{ \Pi(S \times \mathbf{B}K) }} \,, \mathbf{B}G \big) \\ & \simeq \infty Groupoids\left( \mathbf{B} K, \mathbf{B}G \right) \\ & \simeq \mathbf{H}_{sing}\big( \mathbb{B}K, \mathbb{B}G \big) \\ & \simeq \mathbf{H}_{sing}\big( S \times \mathbb{B}K, \mathbb{B}G \big) \end{aligned}

Here we used the various (∞,1)-adjunctions and the (∞,1)-Yoneda lemma, and the claim in turn follows by the (∞,1)-Yoneda lemma.

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GG-Equivariant cohesive toposes

In order to speak of GG-equivariant homotopy theory (Def. below) inside globally equivariant homotopy theory (Prop. above) we need a certain concept of faithfulness (Def. below).

For that purpose, recall that in an (∞,1)-topos the pair of classes of n-connected morphisms and n-truncated morphisms for an orthogonal factorization system for all n{2,1}{}n \in \{-2,-1\} \sqcup \mathbb{N} \sqcup \{\infty\}.

In particular this says that a 1-morphism in an (∞,1)-topos is 0-truncated precisely if it has the right lifting property against every morphism that is 0-connected.

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Just for the record:

Lemma

(Γ sing\Gamma_{sing} preserves n-truncated morphisms)

Let H\mathbf{H} be a cohesive (∞,1)-topos with H sing\mathbf{H}_{sing} its globally equivariant homotopy theory from Prop. .

Then in particular the functor (1)

Γ sing:H singH \Gamma_{sing} \;\colon\; \mathbf{H}_{sing} \longrightarrow \mathbf{H}

preserves n-truncated morphisms for all nn.

Proof

By the (n-connected, n-truncated) factorization system and the adjunctions in (1) the statement is equivalent to

Disc sing:HH sing Disc_{sing} \;\colon\; \mathbf{H} \longrightarrow \mathbf{H}_{sing}

preserving n-connected morphisms. These are effective epimorphisms in an (∞,1)-category satisfying an extra condition. Both the definition of effective epimorphisms as well as that extra conditions are entirely formulated in terms of (∞,1)-limits and (∞,1)-colimits (this Prop.). Since Disc singDisc_{sing} is both a left and a right adjoint (∞,1)-functor by (1) it preserves all these.

Example

(on groupoids (0-connected, 0-truncated) is (eso and full, faithful))

In the (∞,1)-topos ∞Groupoids the 1-truncated objects are equivalently groupoids in the sense of small categories with all morphisms invertible

GroupoidsGroupoids 1Groupoids Groupoids \simeq \infty Groupoids_{\leq 1} \hookrightarrow \infty Groupoids

Under this identification the 1-morphisms between 1-truncated objects correspond equivalently functors, and we have that these 1-morphisms are

  1. 0-truncated precisely if they correspond to faithful functors;

  2. 0-connected precisely if they correspond to essentially surjective and full functors.

In particular a morphism of delooping groupoids

(3)BG Bp BG \array{ \mathbf{B} G' \\ \downarrow^{\mathrlap{\mathbf{B} p}} \\ \mathbf{B} G }

is a 0-connected morphism in Groupoids\infty Groupoids precisely if the corresponding group homomorphism p:GGp \colon G' \to G is surjective.

Therefore one might say “faithful morphism” for every 0-truncated morphism in an (∞,1)-topos. But the terminology “faithful” is used with other meanings, too, and we need to refer to these variants

Definition

(SingularitiesSingularities-faithful morphisms)

Let H\mathbf{H} be a cohesive (∞,1)-topos with H singSh (Singularities,H)\mathbf{H}_{sing} \coloneqq Sh_\infty\left( Singularities, \mathbf{H}\right) its globally equivariant homotopy theory according to Prop. . We say that a morphism 𝒳f𝒴\mathcal{X} \overset{f}{\to} \mathcal{Y} in H sing\mathbf{H}_{sing} is SingularitiesSingularities-faithful if it has the right lifting property against morphisms of the form

(4)(𝔹G 𝔹p 𝔹G 𝔹 p)SingularitiesYonedaSh (Singularities,Groupoids)DiscSh (Singularities,H)=H sing \left( \array{ \mathbb{B}G' \\ \downarrow^{\mathrlap{\mathbb{B}p}} \\ \mathbb{B}G } \phantom{{}^{\mathbb{B}^p}} \right) \;\in\; Singularities \overset{Yoneda}{\hookrightarrow} Sh_\infty\left( Singularities, \infty Groupoids \right) \overset{Disc}{\hookrightarrow} Sh_\infty\left( Singularities, \mathbf{H} \right) \;=\; \mathbf{H}_{sing}

where p:GGp \;\colon\; G' \to G is a surjective group homomorphism.

For the case H=\mathbf{H} = ∞Groupoids this is the definition of faithful maps in Rezk 14. Prop. 3.4.1.

It seems that the morphisms (4) are not 0-connected in Sh (Singularities,Groupoids)Sh_\infty(Singularities, \infty Groupoids). Them being 0-connected should come down to the statement that for p:GGp \colon G' \to G a surjective group homomorphism and HGH \subset G any subgroup, there always is a lift of HH to GG' and that any two such lifts are conjugate to each other, in GG'. But already the first condition fails in general, since not every epimorphism of groups is a split epimorphism.

Nevertheless and in any case we have the following, which is all we will need:

Proposition

(coDisc singcoDisc_{sing} of a 0-truncated morphism is SingularitiesSingularities-faithful)

Let H\mathbf{H} be a cohesive (∞,1)-topos with H sing\mathbf{H}_{sing} its globally equivariant homotopy theory according to Prop. .

If a morphism XfYX \overset{f}{\to} Y in H\mathbf{H} is 0-truncated, then its image under coDisc sing:HH singcoDisc_{sing} \;\colon\; \mathbf{H} \hookrightarrow \mathbf{H}_{sing} is SingularitiesSingularities-faithful (Def. ).

Proof

This follows by the adjunctions (1), the relations (2) and the fact (3):

𝔹p𝔹G coDisc sing(X) 𝔹p coDisc sing(f) 𝔹G coDisc sing(Y)AAAAA 𝔹pΓ sing(𝔹G) X Γ sing(𝔹p) f Γ sing(𝔹G) YAAABG X Bp f BG Y \phantom{{}^{\mathbb{B}p}} \array{ \mathbb{B}G' &\longrightarrow& coDisc_{sing}(X) \\ {}^{\mathllap{\mathbb{B}p}}\big\downarrow &\nearrow& \big \downarrow^{\mathrlap{ coDisc_{sing}(f) }} \\ \mathbb{B}G &\longrightarrow& coDisc_{sing}(Y) } \phantom{AAAA} \;\Leftrightarrow\;\; \phantom{A} \phantom{{}^{\mathbb{B}p}} \array{ \Gamma_{sing}\left(\mathbb{B}G'\right) &\longrightarrow& X \\ {}^{\mathllap{\Gamma_{sing}\left(\mathbb{B}p\right)}}\big\downarrow &\nearrow& \big \downarrow^{\mathrlap{ f }} \\ \Gamma_{sing}\left(\mathbb{B}G\right) &\longrightarrow& Y } \phantom{AA} \;\;\simeq\;\; \phantom{A} \array{ \mathbf{B} G' &\longrightarrow& X \\ {}^{\mathllap{\mathbf{B}p}}\big\downarrow &\nearrow& \big \downarrow^{\mathrlap{ f }} \\ \mathbf{B} G &\longrightarrow& Y }
Definition

(global orbit category)

Write

GlobalOrbitsSingularities faith GlobalOrbits \;\coloneqq\; Singularities^{faith}

for the wide non-full sub-(infinity,1)-category of SingularitiesSingularities (Def. ) with the same objects 𝔹G\mathbb{B}G but the 1-morphisms required to be 0-truncated as morphisms of \infty-groupods, hence to be faithful functors of groupoids (Example ), hence to come from injective group homomorphisms.

Remark

The category GlobalOrbitsGlobalOrbits in Def. , when generalized from finite groups to compact Lie groups, is called

and is not the category called “OrbOrb” in Körschgen 16, Schwede 17, Schwede 18 (see Remark ).

Proposition/Definition

(slice of global orbits is GG-orbits)

For GG a finite group, the slice of the global orbit category from Def. over the object 𝔹G\mathbb{B}G is equivalent to the GG-orbit category

GlobalOrbits /𝔹GGOrbits GlobalOrbits_{/\mathbb{B}G} \;\simeq\; G Orbits
Definition

(GG-equivariant homotopy theory of a cohesive (∞,1)-topos)

For H\mathbf{H} a cohesive (∞,1)-topos and GG a finite group we say that the GG-equivariant homotopy theory of H\mathbf{H} is the (∞,1)-presheaf (∞,1)-topos on the GG-orbit category (Def. ) over H\mathbf{H}:

H G Sh (GOrbits,H) Sh (GlobalOrbits /𝔹G,H) Sh (GlobalOrbits,H) /𝔹G \begin{aligned} \mathbf{H}_G &\coloneqq\; Sh_\infty\big( G Orbits , \mathbf{H} \big) \\ & \simeq Sh_\infty\big( GlobalOrbits_{/\mathbb{B}G} , \mathbf{H} \big) \\ & \simeq Sh_\infty\big( GlobalOrbits , \mathbf{H} \big)_{/\mathbb{B}G} \end{aligned}

On the right we are displaying immediate equivalences, the first by Prop. , the second by the general slicing behaviour of \infty-toposes (this Prop.-topos#SlicingCommutesWithPassingToPresheaves)).

The relation between the global homotopy theory H sing\mathbf{H}_{sing} (Prop. ) and the GG-equivariant homotopy theory H G\mathbf{H}_G (Def. ) is the topic of Rezk 14:

Definition

(normal subgroup classifier, Rezk 14, 4.1)

For H\mathbf{H} a cohesive (∞,1)-topos, let

𝒩Sh(Singularities,Sets)Sh (Singularities,Groupoids)Sh (Singularities,H) \mathcal{N} \;\in\; Sh\big(Singularities, Sets \big) \hookrightarrow Sh_\infty\big(Singularities, \infty Groupoids \big) \overset{}{\hookrightarrow} Sh_\infty\big( Singularities, \mathbf{H} \big)

be the presheaf which to any finite group 𝔹G\mathbb{B}G assigns the set (i.e. 0-groupoid) of normal subgroups of GG.

Proposition

(Rezk 14)

For H\mathbf{H} an (∞,1)-topos, with H singSh (Singularities,)\mathbf{H}_{sing} \coloneqq Sh_\infty\big( Singularities, \mathbf{}\big) its global equivariant homotopy theory according to Prop. , there is an equivalence of (∞,1)-categories

Sh (GlobalOrbits,H) (Sh (Singularities,H) /𝒩) singfaith=((H sing) /𝒩) singfaith \array{ Sh_\infty\big( GlobalOrbits, \mathbf{H}\big) &\overset{\simeq}{\longrightarrow}& \left( Sh_\infty\big( Singularities, \mathbf{H}\big)_{/\mathcal{N}} \right)_{\text{singfaith}} \;=\; \left( \left( \mathbf{H}_{sing} \right)_{/\mathcal{N}} \right)_{singfaith} }

between the (∞,1)-presheaf (∞,1)-category on the global orbit category according to Def. and the full sub-(∞,1)-category of the slice (∞,1)-topos of H sing\mathbf{H}_{sing} over the normal subgroup classifier 𝒩\mathcal{N} (Def. ) on those morphisms to 𝒩\mathcal{N} whhich are SingularitiesSingularities-faithful according to Def. .

Moreover, if GG is a finite group then slicing over 𝔹G\mathbb{B}G this yields an equivalence

H G((H sing) /𝔹G) singfaith \array{ \mathbf{H}_G \;\simeq\; \left( \left( \mathbf{H}_{sing} \right)_{/\mathbb{B}G} \right)_{singfaith} }

between the GG-equivariant homotopy theory H G\mathbf{H}_G (Def. ) and the full sub-(∞,1)-category on the SingularitiesSingularities-faithful objects of the slice of the global homotopy theory H sing\mathbf{H}_{sing} (Prop. ) over 𝔹G\mathbb{B}G.

Orbifolds

With cohesive globally equivariant homotopy theory in place, there is now an elegant definition of orbifolds (Def. below) which, being fully synthetic makes manifest good defining properties of the category of orbifolds (Remark below).

Not directly evident is that under passing to shapes (underlying globally equivariant geometrically discrete ∞-groupoids) this definition is compatible with the standard embedding of G-spaces into globally equivariant homotopy theory. That this indeed is the case is confirmed in Example below.

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Definition

(orbifold)

Let H\mathbf{H} be a cohesive (∞,1)-topos with H sing\mathbf{H}_{sing} its corresponding globally equivariant homotopy theory according to Prop. .

For GGrp(H)G \in Grp(\mathbf{H}) a finite group we say that an orbifold with singularities (isotropy groups) in GG is an object of the slice (∞,1)-topos

𝒳(H sing) /𝔹G \mathcal{X} \;\in\; \left(\mathbf{H}_{sing}\right)_{/\mathbb{B}G}

which is

  1. 0-truncated (as an object of the slice);

  2. sing\sharp_{sing}-modal (hence “SingularitySingularity-codiscrete”).

Given such an orbifold, we say that its underlying geometric groupoid is its SingularitiesSingularities-flat aspect:

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

where on the right we used (2).

If H\mathbf{H} is moreover differentially cohesive and VGrp(H)V \in Grp(\mathbf{H}) is a group object, then an orbifold 𝒳\mathcal{X} is called a VV-orbifold if its underlying geometric groupoid sing(𝒳)\flat_{sing}\left( \mathcal{X}\right) (5) is a V-manifold.

Example

(global homotopy quotient-orbifolds)

Let XHX \in \mathbf{H} be 0-truncated and equipped with a GG-action, with homotopy quotient (XBGBG)H /BG(X \sslash \mathbf{B}G \to \mathbf{B}G) \in \mathbf{H}_{/\mathbf{B}G}. Then

𝒳coDisc sing(XG BG)(coDisc sing(XG) 𝔹G) \mathcal{X} \;\coloneqq\; coDisc_{sing} \left( \array{ X \sslash G \\ \downarrow \\ \mathbf{B}G } \right) \;\simeq\; \left( \array{ coDisc_{sing}\left( X\sslash G\right) \\ \downarrow \\ \mathbb{B}G } \right)

is an orbifold with isotropy groups in GG, according to Def. . Here on the right we identified the slice using Prop. .

As a further special case:

Example

(global homotopy quotient-orbifolds of smooth manifolds)

Let H\mathbf{H} \coloneqq Smooth∞Groupoids. For GG a finite group, let XX be a smooth manifold equipped with a smooth GG-action. Under the canonical embedding into H\mathbf{H} the corresponding action groupoid is a 0-truncated object

(XG BG)SmoothGroupoids /BG \left( \array{ X \sslash G \\ \downarrow \\ \mathbf{B}G } \right) \;\in\; Smooth\infty Groupoids_{/\mathbf{B}G}

as in Example , and hence its SingularitiesSingularities-codiscrete image is a orbifold in the sense of Def. :

𝒳(coDisc sing(XG) 𝔹G)(SmoothGroupoids sing) /𝔹G. \mathcal{X} \;\coloneqq\; \left( \array{ coDisc_{sing}\big(X \sslash G \big) \\ \downarrow \\ \mathbb{B}G } \right) \;\in\; \left( Smooth\infty Groupoids_{sing} \right)_{/\mathbb{B}G} \,.

We claim that the shape of this orbifold in plain global equivariant homotopy theory coincides with the global equivariant homotopy type associated with the G-space underlying XX

ʃcoDisc sing(XG)Δ G(X)Sh (Singularities,Groupoids), ʃ coDisc_{sing}\left( X \sslash G \right) \;\simeq\; \Delta_G\big( X \big) \;\in\; Sh_\infty\big( Singularities, \infty Groupoids \big) \,,

where on the right Δ G\Delta_G is as in Rezk 14, 3.2

Proof

The key point is that the assumption of 0-truncation of 𝒳\mathcal{X} and the restriction to finite (hence discrete) groups ensures that coDisc singcoDisc_{sing} forms the correct fixed point sheaves, whose separate shape/geometric realization then coincides with the relevant fixed point spaces of XX.

In detail, as \infty-groupoid-valued presheaves on the product site CartSp ×Singularities\times Singularities we have

coDisc sing(XG): n×𝔹K Groupoids(BK,C ( n,X)Set1GroupoidG) 1GroupoidsGrpd(Set)(BK,C ( n,X)G) ϕGroups(K,G)C ( n,X ϕ(K))G \begin{aligned} coDisc_{sing}\big( X\sslash G\big) \;\colon\; \mathbb{R}^n \times \mathbb{B}K & \mapsto\; \infty Groupoids\big( \mathbf{B}K, \underset{ \in 1Groupoid }{ \underbrace{ \underset{\in Set}{ \underbrace{C^\infty(\mathbb{R}^n, X)} } } } \sslash G \big) \\ & \simeq \; \underset{ \simeq Grpd(Set) }{ \underbrace{ 1 Groupoids } } \big( \mathbf{B}K, C^\infty(\mathbb{R}^n, X) \sslash G \big) \\ & \simeq \; \underset{\phi \in Groups(K,G)}{\bigsqcup} C^\infty(\mathbb{R}^n, X^{\phi(K)}) \sslash G \end{aligned}

where in the first step we used the adjunction (Γ singcoDisc sing)(\Gamma_{sing} \dashv coDisc_{sing}) as in Prop. . Hence as smooth \infty-groupoid valued presheaves on just SingularitiesSingularities this is

(coDisc sing(XG):𝔹KϕGroups(K,G)X ϕ(K)G)Sh (Singularities,SmoothGroupoids). \Big( coDisc_{sing}\big( X \sslash G \big) \;\colon\; \mathbb{B}K \;\mapsto\; \underset{\phi \in Groups(K,G)}{\bigsqcup} X^{\phi(K)} \sslash G \Big) \;\in\; Sh_\infty\big( Singularities, Smooth\infty Groupoids \big) \,.

Now shape ʃ is left adjoint, hence preserves the coproducts and the homotopy quotient by GG and finally als GG itself (GG being discrete, and ʃ preserving the point (the terminal object), by cohesion), so that in conclusion

(ʃcoDisc sing(XG):𝔹KϕGroups(K,G)ʃ(X ϕ(K))G)Sh (Singularities,Groupoids). \Big( ʃ coDisc_{sing}\big( X \sslash G \big) \;\colon\; \mathbb{B}K \;\mapsto\; \underset{\phi \in Groups(K,G)}{\bigsqcup} ʃ \left(X^{\phi(K)}\right) \sslash G \Big) \;\in\; Sh_\infty\big( Singularities, \infty Groupoids \big) \,.

But this is exactly the formula for Δ G(X)\Delta_G (X), as in Rezk 14, 3.2.

Proposition

(orbifolds are in cohesive equivariant homotopy theory)

An orbifold 𝒳\mathcal{X} with isotropy groups in GG, according to Def. is SingularitiesSingularities-faithful over 𝔹G\mathbb{B}G (Def. ) and hence inside the inclusion (from Prop. ) of the GG-equivariant homotopy theory of H\mathbf{H} (Def. ) into the globally equivariant homotopy theory of H\mathbf{H}:

𝒳H G(H sing) /𝔹G \mathcal{X} \;\in\; \mathbf{H}_G \;\hookrightarrow\; \left( \mathbf{H}_{sing} \right)_{/\mathbb{B}G}
Proof

By Prop. we need to check that 𝒳𝔹G\mathcal{X} \to \mathbb{B}G is SingularitiesSingularities-faithful.

Now, by the first defining assumption on 𝒳\mathcal{X} (Def. ) and by Lemma , we have that Γ sing(𝒳𝔹G)\Gamma_{sing}(\mathcal{X} \to \mathbb{B}G) is 0-truncated. By cohesion (1) we have coDisc singΓ singIdcoDisc_{sing} \circ \Gamma_{sing} \;\simeq\; Id and hence 𝒳𝔹G\mathcal{X} \to \mathbb{B}G is the image under coDisc singcoDisc_{sing} of a 0-truncated morphism. With this the statement follows by Prop. .

Remark

(orbifolds inside globally equivariant homotopy theory are still equivalent to cohesive groupoids)

Since coDisc sing:HH singcoDisc_{sing} \;\colon\; \mathbf{H} \hookrightarrow \mathbf{H}_{sing} is a full sub-(∞,1)-category-inclusion, the (2,1)-category of VV-orbifolds inside H sing\mathbf{H}_{sing} according to Def. is equivalent to its pre-image in H\mathbf{H}, hence will coincide, for suitable choices of H\mathbf{H} and VGrp(H)V \in Grp(\mathbf{H}), with traditional definition of (2,1)-categories of orbifolds regarded as certain geometric groupoids. But by embedding this into the larger global homotopy theory H sing\mathbf{H}_{sing} of H\mathbf{H} more general coefficient-objects for orbifold cohomology become available, and this brings in the previously missing Bredon-equivariant cohomology-aspect of orbifold cohomology.

\,

Orbifold cohomology

With orbifolds properly realized in cohesive globally equivariant homotopy theory by the above, the proper definition of orbifold cohomology is now immediate, by the general logic of cohomology in (∞,1)-toposes, this is Def. below.

What needs checking is that this definition, in the special case that the coefficient object is geometrically discrete reproduces Bredon equivariant cohomology of the underlying bare homotopy type of the orbifold. But this follows readily with the general considerations above, this is Example below.

\,

Definition

(orbifold cohomology)

Let H\mathbf{H} be a cohesive (∞,1)-topos and write H sing\mathbf{H}_{sing} for its globally equivariant homotopy theory as in Prop. .

Let GG be a finite group and consider an orbifold with isotropy groups/singularities in GG, according to Def. :

𝒳(H sing). \mathcal{X} \;\in\; \left( \mathbf{H}_{sing} \right) \,.

Then for

𝒜(H sing) \mathcal{A} \;\in\; \left( \mathbf{H}_{sing} \right)

any other object (not necessarily itself an orbifold, and typically far from being so) the orbifold cohomology of 𝒳\mathcal{X} with coefficients in 𝒜\mathcal{A} is the cohomology as given by the ambient (∞,1)-topos, hence

H(𝒳,𝒜)(H sing) /𝔹G(𝒳,𝒜). H\big( \mathcal{X}, \mathcal{A} \big) \;\coloneqq\; \left( \mathbf{H}_{sing} \right)_{/\mathbb{B}G} \big( \mathcal{X}, \mathcal{A}\big) \,.

We check the two desiderata for a good definition of orbifold cohomology discussed above:

  1. It is clear that on objects in the inclusion Disc sing:HH singDisc_{sing} \;\colon\; \mathbf{H} \hookrightarrow \mathbf{H}_{sing} orbifold cohomology reduces to geometric cohomology on H\mathbf{H}.

  2. In the other extreme, for orbifold cohomology with geometrically discrete coefficients, we check that we re-obtain Bredon equivariant cohomology of the underlying G-spaces:

Example

(orbifold cohomology with geometrically discrete coefficients is GG-equivariant cohomology)

Let H=\mathbf{H} = Smooth∞Groupoids and consider a smooth manifold XX equipped with smooth action by a finite group GG, regarded as an orbifold as in Example :

𝒳(coDisc sing(XG) 𝔹G)(SmoothGroupoids sing) /𝔹G. \mathcal{X} \;\coloneqq\; \left( \array{ coDisc_{sing}\big( X \sslash G \big) \\ \downarrow \\ \mathbb{B}G } \right) \;\in\; \Big( Smooth \infty Groupoids_{sing} \Big)_{/\mathbb{B}G} \,.

Let moreover

𝒜(Groupoids) GDisc(SmoothGroupoids) G(SmoothGroupoids sing) /𝔹G \mathcal{A} \;\in\; \Big( \infty Groupoids \Big)_{G} \overset{Disc}{\hookrightarrow} \Big( Smooth \infty Groupoids \Big)_{G} \hookrightarrow \Big( Smooth \infty Groupoids_{sing} \Big)_{/\mathbb{B}G}

be any geometrically discrete coefficient object in GG-equivariant homotopy theory, included into the slice of the global equivariant homotopy theory via Prop. .

Then the orbifold cohomology of 𝒳\mathcal{X} with coefficients in 𝒜\mathcal{A}, according to Def. , coincides with the Bredon GG-equivariant cohomology of the G-space underlying XX:

H(𝒳,Disc(𝒜))H G(X,𝒜). H\big( \mathcal{X}, Disc(\mathcal{A}) \big) \;\simeq\; H_G\big( X, \mathcal{A} \big) \,.
Proof

We compute

(SmoothGroupoids sing) /𝔹G(𝒳,Disc(𝒜)) SmoothGroupoids G(𝒳,Disc(𝒜)) Groupoids G(ʃ sing𝒳,𝒜) \begin{aligned} \left(Smooth \infty Groupoids_{sing}\right)_{/\mathbb{B}G} \left( \mathcal{X}, Disc(\mathcal{A}) \right) & \simeq Smooth \infty Groupoids_G \left( \mathcal{X}, Disc(\mathcal{A}) \right) \\ & \simeq \infty Groupoids_G \left( ʃ_{sing}\mathcal{X}, \mathcal{A} \right) \end{aligned}

where the first step is by Prop. , while the second is by the cohesion adjunction ʃDiscʃ \dashv Disc for Smooth∞Groupoids. By Example we have that ʃ sing𝒳Groupoids Gʃ_{sing}\mathcal{X} \in \infty Groupoids_G is indeed the G-space XX regarded in GG-equivariant homotopy theory.

History

According to Abramovich 05, p. 42:

On December 7, 1995 Maxim Kontsevich delivered a history-making lecture at Orsay, titled String Cohomology. This is what is now know, after Chen-Ruan 00, as orbifold cohomology, Kontsevich’s lecture notes described the orbifold and quantum cohomology of a global quotient orbifold. Twisted sectors, the age grading, and a version of orbifold stable maps for global quotients are all there.

The same lecture also introduced motivic integration.

References

Discussion of orbifold cohomology in the context of Bredon/global equivariant cohomology includes

related to results of

Discussion of cohomology of inertia orbifolds is due to

See also

Last revised on December 2, 2018 at 18:24:02. See the history of this page for a list of all contributions to it.