group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
function algebras on ∞-stacks?
derived smooth geometry
Since the crucial extra structures carried by an orbifold are
geometric structure (e.g. topological, algebro-geometric, differential-geometric, super-geometric, etc.)
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
geometric cohomology (e.g. sheaf hypercohomology, differential cohomology, etc.)
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).
$\,$
A proposal popular among Lie theorists (Moerdijk-Pronk 97) is to regard an orbifold with local charts $U_i \in G_i Actions$ (actions of some group on some local model space) as the geometric stack obtained by gluing the corresponding homotopy quotients/quotient stacks $U_i \!\sslash \!G_i$. If $\mathbf{H}$ is the ambient cohesive (∞,1)-topos in which this takes place (for instance $\mathbf{H} =$ Smooth∞Groupoids, SuperFormalSmooth∞Groupoids, etc.) and if $G \in Grp \overset{Disc}{\hookrightarrow} Grp(\mathbf{H})$ is a discrete group in which all the isotropy groups of the orbifold are contained, this gives an object
in the slice (∞,1)-topos over the delooping $\mathbf{B}G = \ast \sslash G$ of $G$, which is still a 0-truncated object, reflecting that as a functor of groupoids the morphism $\mathcal{X} \to \mathbf{B}G$ is a faithful functor.
Accordingly, if this is – or were – the correct formalization of the nature of orbifolds $\mathcal{X}$, then the corresponding orbifold cohomology has coefficients given by objects $\mathcal{A} \in \mathbf{H}_{/\mathbf{B}G}$ and cohomology sets being the connected components of the (∞,1)-categorical hom-spaces
This concept of orbifold cohomology does fully reflect the geometric nature of orbifolds. It also reflects some equivariance aspect. For example if $\mathcal{X} = \ast \sslash G$ is the one-point orbifold with singularity given by a finite group $G$, and if $V \in G Representations$ is a linear representation, with $K(V,n)\sslash G \in \infty Groupoids_{/\mathbf{B}G} \overset{Disc}{\hookrightarrow} \mathbf{H}_{/\mathbf{B}G}$ its Eilenberg-MacLane space, then
is the group cohomology of $G$ with coefficients in $V$.
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.
$\,$
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 $\mathbb{B}G$.
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 $\ast = \ast / G$.
The Lie geometer says that what she sees is a point transforming under the $G$-action that fixes it, hence the homotopy quotient groupoid $\mathbf{B}G =\ast \sslash G$.
These are two opposite extreme aspects of the orbifold singularity $\mathbb{B}G$, 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 $\mathbf{B} G$ (recall that already $\mathbf{B}\mathbb{Z}_2 = \mathbb{R}P^\infty$), but it is a point that also remembers the group action, for that characterizes how the singularity is being singular.
The definition of orbifold cohomology below in Def. is the canonical cohomology in a slice of the globally equivariant homotopy theory $\mathbf{H}_{sing}$ of the given ambient cohesive (∞,1)-topos $\mathbf{H}$.
For completeness, we first introduce/recall $\mathbf{H}_{sing}$ in Prop. below, as well as its slices to $G$-equivariant homotopy theory $\mathbf{H}_G$ (Def. below), following Rezk 14.
The key point in the following is the identification of orbifolds as the 0-truncated “$Singularities$-codiscrete objects” in $\mathbf{H}_{sing}$ (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.)
$\,$
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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(category of singularities)
Write
for the (2,1)-category of connected finite groupoids. A skeleton has objects labeled by finite groups $G$, and we will denote these objects
to distinguish them from their image as delooping groupoids $\mathbf{B} G \in$ ∞Grpd. (As we consider (∞,1)-presheaves on $Singularities$ with values in ∞Groupoids, in Prop. below, these two objects become crucially different, albeit closely related.)
The category $Singularities$ in Def. , when generalized from finite groups to compact Lie groups, is called
“$Orb$ version 1” in Henriques-Gepner 07
“$Glo$” in Rezk 14, 2.2
“$Orb$” in Körschgen 16, Schwede 17, Schwede 18.
(global equivariant homotopy theory cohesive over base (∞,1)-topos)
Let $\mathbf{H}$ be any (∞,1)-topos and consider the (∞,1)-category of (∞,1)-presheaves on the category of singularities (Def. ) over the base (∞,1)-topos $\mathbf{H}$, hence the (∞,1)-functor (∞,1)-category
This is a cohesive (∞,1)-topos over the base (∞,1)-topos $\mathbf{H}$ in that the global section-geometric morphisms enhances to an adjoint quadruple of adjoint (∞,1)-functors
such that
$Disc_{sing}, coDisc_{sing} \;\colon\; \mathbf{H} \to \mathbf{H}_{sing}$ are fully faithful (∞,1)-functors;
$\Pi_{sing}$ preserves finite products.
hence inducing an adjoint triple of adjoint modalities
(“shape”, “flat”, “sharp” for singularities).
Moreover, for $G$ a finite group regarded under the inclusion
and writing
for its delooping under $Grp\left( \mathbf{H}_{sing} \right) \underoverset{\simeq}{\mathbf{B}}{\longrightarrow} \mathbf{H}^{\ast/}_{cn}$,
in constrast to the (∞,1)-Yoneda embedding
we have
This is immediate by general properties of left/right (∞,1)-Kan extension, using the evident fact that $Singularities$ has finite products (the terminal object is $\mathbb{B}1$ and the binary Cartesian product is give by forming direct product groups: $\left(\mathbb{B}G_1\right) \times \left( \mathbb{B}G_2\right) \simeq \mathbb{B}\left( G_1 \times G_2\right)$ ). The directly analogous 1-categorical argument is at infinity-cohesive site.
For $\mathbf{H} =$ ∞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:
(orbifold singularities are the codiscrete aspect of homotopy quotients)
Let $\mathbf{H}$ itself be a cohesive (∞,1)-topos over ∞Groupoids
Then in the situation of Prop.
Let $\mathcal{S}$ be a cohesive (∞,1)-site of definition for $\mathbf{H}$, so that
and $\Pi(S) \simeq \ast$ for $S \in \mathcal{S} \overset{y}{\hookrightarrow} \mathbf{H}$.
Then as (∞,1)-presheaves regarded this way we have
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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In order to speak of $G$-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 $n \in \{-2,-1\} \sqcup \mathbb{N} \sqcup \{\infty\}$.
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.
$\,$
Just for the record:
($\Gamma_{sing}$ preserves n-truncated morphisms)
Let $\mathbf{H}$ be a cohesive (∞,1)-topos with $\mathbf{H}_{sing}$ its globally equivariant homotopy theory from Prop. .
Then in particular the functor (1)
preserves n-truncated morphisms for all $n$.
By the (n-connected, n-truncated) factorization system and the adjunctions in (1) the statement is equivalent to
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 $Disc_{sing}$ is both a left and a right adjoint (∞,1)-functor by (1) it preserves all these.
(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
Under this identification the 1-morphisms between 1-truncated objects correspond equivalently functors, and we have that these 1-morphisms are
0-truncated precisely if they correspond to faithful functors;
0-connected precisely if they correspond to essentially surjective and full functors.
In particular a morphism of delooping groupoids
is a 0-connected morphism in $\infty Groupoids$ precisely if the corresponding group homomorphism $p \colon G' \to G$ 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
($Singularities$-faithful morphisms)
Let $\mathbf{H}$ be a cohesive (∞,1)-topos with $\mathbf{H}_{sing} \coloneqq Sh_\infty\left( Singularities, \mathbf{H}\right)$ its globally equivariant homotopy theory according to Prop. . We say that a morphism $\mathcal{X} \overset{f}{\to} \mathcal{Y}$ in $\mathbf{H}_{sing}$ is $Singularities$-faithful if it has the right lifting property against morphisms of the form
where $p \;\colon\; G' \to G$ is a surjective group homomorphism.
For the case $\mathbf{H} =$ ∞Groupoids this is the definition of faithful maps in Rezk 14. Prop. 3.4.1.
It seems that the morphisms (4) are not in general 0-connected in $Sh_\infty(Singularities, \infty Groupoids)$. Them being 0-connected should come down to the statement that for $p \colon G' \to G$ a surjective group homomorphism and $H \subset G$ any subgroup, there always is a lift of $H$ to $G'$ and that any two such lifts are conjugate to each other, in $G'$. 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:
($coDisc_{sing}$ of a 0-truncated morphism is $Singularities$-faithful)
Let $\mathbf{H}$ be a cohesive (∞,1)-topos with $\mathbf{H}_{sing}$ its globally equivariant homotopy theory according to Prop. .
If a morphism $X \overset{f}{\to} Y$ in $\mathbf{H}$ is 0-truncated, then its image under $coDisc_{sing} \;\colon\; \mathbf{H} \hookrightarrow \mathbf{H}_{sing}$ is $Singularities$-faithful (Def. ).
This follows by the adjunctions (1), the relations (2) and the fact (3):
Write
for the wide non-full sub-(infinity,1)-category of $Singularities$ (Def. ) with the same objects $\mathbb{B}G$ 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.
The category $GlobalOrbits$ in Def. , when generalized from finite groups to compact Lie groups, is called
“$Orb$ version 2” in Henriques-Gepner 07
“$Orb$” in Rezk 14, 4.5
and is not the category called “$Orb$” in Körschgen 16, Schwede 17, Schwede 18 (see Remark ).
(slice of global orbits is $G$-orbits)
For $G$ a finite group, the slice of the global orbit category from Def. over the object $\mathbb{B}G$ is equivalent to the $G$-orbit category
($G$-equivariant homotopy theory of a cohesive (∞,1)-topos)
For $\mathbf{H}$ a cohesive (∞,1)-topos and $G$ a finite group we say that the $G$-equivariant homotopy theory of $\mathbf{H}$ is the (∞,1)-presheaf (∞,1)-topos on the $G$-orbit category (Def. ) over $\mathbf{H}$:
On the right we are displaying immediate equivalences, the first by Prop. , the second by the general slicing behaviour of $\infty$-toposes (this Prop.).
The relation between the global homotopy theory $\mathbf{H}_{sing}$ (Prop. ) and the $G$-equivariant homotopy theory $\mathbf{H}_G$ (Def. ) is the topic of Rezk 14:
(normal subgroup classifier, Rezk 14, 4.1)
For $\mathbf{H}$ a cohesive (∞,1)-topos, let
be the presheaf which to any finite group $\mathbb{B}G$ assigns the set (i.e. 0-groupoid) of normal subgroups of $G$.
(Rezk 14)
For $\mathbf{H}$ an (∞,1)-topos, with $\mathbf{H}_{sing} \coloneqq Sh_\infty\big( Singularities, \mathbf{H}\big)$ its global equivariant homotopy theory according to Prop. , there is an equivalence of (∞,1)-categories
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 $\mathbf{H}_{sing}$ over the normal subgroup classifier $\mathcal{N}$ (Def. ) on those morphisms to $\mathcal{N}$ which are $Singularities$-faithful according to Def. .
Moreover, if $G$ is a finite group then slicing over $\mathbb{B}G$ this yields an equivalence
between the $G$-equivariant homotopy theory $\mathbf{H}_G$ (Def. ) and the full sub-(∞,1)-category on the $Singularities$-faithful objects of the slice of the global homotopy theory $\mathbf{H}_{sing}$ (Prop. ) over $\mathbb{B}G$.
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.
$\,$
(orbifold)
Let $\mathbf{H}$ be a cohesive (∞,1)-topos with $\mathbf{H}_{sing}$ its corresponding globally equivariant homotopy theory according to Prop. .
For $G \in Grp(\mathbf{H})$ a finite group we say that an orbifold with singularities (isotropy groups) in $G$ is an object of the slice (∞,1)-topos
which is
0-truncated (as an object of the slice);
$\sharp_{sing}$-modal (hence “$Singularity$-codiscrete”).
Given such an orbifold, we say that its underlying geometric groupoid is its $Singularities$-flat aspect:
where on the right we used (2).
If $\mathbf{H}$ is moreover differentially cohesive and $V \in Grp(\mathbf{H})$ is a group object, then an orbifold $\mathcal{X}$ is called a $V$-orbifold if its underlying geometric groupoid $\flat_{sing}\left( \mathcal{X}\right)$ (5) is a V-manifold.
(global homotopy quotient-orbifolds)
Let $X \in \mathbf{H}$ be 0-truncated and equipped with a $G$-action, with homotopy quotient $(X \sslash \mathbf{B}G \to \mathbf{B}G) \in \mathbf{H}_{/\mathbf{B}G}$. Then
is an orbifold with isotropy groups in $G$, according to Def. . Here on the right we identified the slice using Prop. .
As a further special case:
(global homotopy quotient-orbifolds of smooth manifolds)
Let $\mathbf{H} \coloneqq$ Smooth∞Groupoids. For $G$ a finite group, let $X$ be a smooth manifold equipped with a smooth $G$-action. Under the canonical embedding into $\mathbf{H}$ the corresponding action groupoid is a 0-truncated object
as in Example , and hence its $Singularities$-codiscrete image is a orbifold in the sense of Def. :
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 $X$
where on the right $\Delta_G$ is as in Rezk 14, 3.2
The key point is that the assumption of 0-truncation of $\mathcal{X}$ and the restriction to finite (hence discrete) groups ensures that $coDisc_{sing}$ forms the correct fixed point sheaves, whose separate shape/geometric realization then coincides with the relevant fixed point spaces of $X$.
In detail, as $\infty$-groupoid-valued presheaves on the product site CartSp $\times Singularities$ we have
where in the first step we used the adjunction $(\Gamma_{sing} \dashv coDisc_{sing})$ as in Prop. . Hence as smooth $\infty$-groupoid valued presheaves on just $Singularities$ this is
Now shape ʃ is left adjoint, hence preserves the coproducts and the homotopy quotient by $G$ and finally also $G$ itself ($G$ being discrete, and ʃ preserving the point (the terminal object), by cohesion), so that in conclusion
But this is exactly the formula for $\Delta_G (X)$, as in Rezk 14, 3.2.
(orbifolds are in cohesive equivariant homotopy theory)
An orbifold $\mathcal{X}$ with isotropy groups in $G$, according to Def. is $Singularities$-faithful over $\mathbb{B}G$ (Def. ) and hence inside the inclusion (from Prop. ) of the $G$-equivariant homotopy theory of $\mathbf{H}$ (Def. ) into the globally equivariant homotopy theory of $\mathbf{H}$:
By Prop. we need to check that $\mathcal{X} \to \mathbb{B}G$ is $Singularities$-faithful.
Now, by the first defining assumption on $\mathcal{X}$ (Def. ) and by Lemma , we have that $\Gamma_{sing}(\mathcal{X} \to \mathbb{B}G)$ is 0-truncated. By cohesion (1) we have $coDisc_{sing} \circ \Gamma_{sing} \;\simeq\; Id$ and hence $\mathcal{X} \to \mathbb{B}G$ is the image under $coDisc_{sing}$ of a 0-truncated morphism. With this the statement follows by Prop. .
(orbifolds inside globally equivariant homotopy theory are still equivalent to cohesive groupoids)
Since $coDisc_{sing} \;\colon\; \mathbf{H} \hookrightarrow \mathbf{H}_{sing}$ is a full sub-(∞,1)-category-inclusion, the (2,1)-category of $V$-orbifolds inside $\mathbf{H}_{sing}$ according to Def. is equivalent to its pre-image in $\mathbf{H}$, hence will coincide, for suitable choices of $\mathbf{H}$ and $V \in Grp(\mathbf{H})$, with traditional definition of (2,1)-categories of orbifolds regarded as certain geometric groupoids. But by embedding this into the larger global homotopy theory $\mathbf{H}_{sing}$ of $\mathbf{H}$ more general coefficient-objects for orbifold cohomology become available, and this brings in the previously missing Bredon-equivariant cohomology-aspect of orbifold cohomology.
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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.
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(orbifold cohomology)
Let $\mathbf{H}$ be a cohesive (∞,1)-topos and write $\mathbf{H}_{sing}$ for its globally equivariant homotopy theory as in Prop. .
Let $G$ be a finite group and consider an orbifold with isotropy groups/singularities in $G$, according to Def. :
Then for
any other object (not necessarily itself an orbifold, and typically far from being so) the orbifold cohomology of $\mathcal{X}$ with coefficients in $\mathcal{A}$ is the cohomology as given by the ambient (∞,1)-topos, hence
We check the two desiderata for a good definition of orbifold cohomology discussed above:
It is clear that on objects in the inclusion $Disc_{sing} \;\colon\; \mathbf{H} \hookrightarrow \mathbf{H}_{sing}$ orbifold cohomology reduces to geometric cohomology on $\mathbf{H}$.
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:
(orbifold cohomology with geometrically discrete coefficients is $G$-equivariant cohomology)
Let $\mathbf{H} =$ Smooth∞Groupoids and consider a smooth manifold $X$ equipped with smooth action by a finite group $G$, regarded as an orbifold as in Example :
Let moreover
be any geometrically discrete coefficient object in $G$-equivariant homotopy theory, included into the slice of the global equivariant homotopy theory via Prop. .
Then the orbifold cohomology of $\mathcal{X}$ with coefficients in $\mathcal{A}$, according to Def. , coincides with the Bredon $G$-equivariant cohomology of the G-space underlying $X$:
We compute
where the first step is by Prop. , while the second is by the cohesion adjunction $ʃ \dashv Disc$ for Smooth∞Groupoids. By Example we have that $ʃ \mathcal{X} \in \infty Groupoids_G$ is indeed the G-space $X$ regarded in $G$-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.
Discussion of orbifold cohomology in the context of Bredon/global equivariant cohomology includes
Dorette Pronk, Laura Scull, Translation Groupoids and Orbifold Bredon Cohomology, Canad. J. Math. 62(2010), 614-645 (arXiv:0705.3249, doi:10.4153/CJM-2010-024-1)
Stefan Schwede, Orbispaces, orthogonal spaces, and the universal compact Lie group (arXiv:1711.06019)
Stefan Schwede, Global homotopy theory, New Mathematical Monographs, 34 Cambridge University Press, 2018 (arXiv:1802.09382)
related to results of
André Henriques, David Gepner, Homotopy Theory of Orbispaces (arXiv:math/0701916)
Discussion of cohomology of inertia orbifolds is due to
See also
Last revised on June 22, 2019 at 12:16:05. See the history of this page for a list of all contributions to it.