nLab orbifold cohomology

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).

Deficiency of orbifolds as geometric groupoids

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

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

in the slice (∞,1)-topos over the delooping $\mathbf{B}G = \ast \sslash G$ 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

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

This concept of orbifold cohomology does fully reflect the geometric nature of orbifolds. It also reflects some equivariance aspect. For example if $\mathcal{X} = \ast \sslash G$ 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

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

is the group cohomology of $G$ 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.

$\,$

Lift of orbifolds to singular-cohesive 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 $\mathbb{B}G$.

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:

$$\array{ && { \text{orbifold singularity} \atop {\mathbb{B}G} } \\ & {}^{\mathllap{\boxed{\lt}}}\swarrow & {{\phantom{A}} \atop { \text{opposite extreme} \atop \text{aspects of orbifold singularity} }} & \searrow^{\mathrlap{ \boxed{\subset} }} \\ { \text{plain quotient} \atop {\ast = \ast/G} } && && { \text{homotopy quotient} \atop { \mathbf{B}G = \ast \sslash G } } }$$

This “unity of opposites” may be captured by the modalities on the singular-cohesive $\infty$-topos of singular-smooth $\infty$-groupoids. Its intrinsic cohomology accomodates a good notion of orbifold cohomology (SaSc 2020).

Related concepts

• orbifold

• Gromov-Witten theory

• orbifold K-theory, equivariant K-theory

• twisted ad-equivariant Tate K-theory

• equivariant elliptic cohomology

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

Traditional orbifold cohomology

Traditionally, the cohomology of orbifolds has, by and large, been taken to be simply the ordinary cohomology of (the plain homotopy type of) the geometric realization of the topological/Lie groupoid corresponding to the orbifold.

For the global quotient orbifold of a G-space $X$, this is the ordinary cohomology of (the bare homotopy type of) the Borel construction $X \!\sslash\! G \;\simeq\; X \times_G E G$, hence is Borel cohomology (as opposed to finer versions of equivariant cohomology such as Bredon cohomology).

A dedicated account of this Borel cohomology of orbifolds, in the generality of twisted cohomology (i.e. with local coefficients) is in:

• Ieke Moerdijk, Dorette Pronk, Simplicial cohomology of orbifolds, Indagationes Mathematicae Volume 10, Issue 2, 1999, Pages 269-293 (doi:10.1016/S0019-3577(99)80021-4)

Moreover, since the orbifold’s isotropy groups $G_x$ are, by definition, finite groups, their classifying spaces $\ast \!\sslash\! G \simeq B G$ have purely torsion integral cohomology in positive degrees, and hence become indistinguishable from the point in rational cohomology (and more generally whenever their order is invertible in the coefficient ring).

Therefore, in the special case of rational/real/complex coefficients, the traditional orbifold Borel cohomology reduces further to an invariant of just (the homotopy type of) the naive quotient underlying an orbifold. For global quotient orbifolds this is the topological quotient space $X/G$.

In this form, as an invariant of just $X/G$, the real/complex/de Rham cohomology of orbifolds was originally introduced in

• Ichiro Satake, On a generalisation of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359–363 (doi:10.1073/pnas.42.6.359)

following analogous constructions in

• Walter Lewis Baily, On the quotient of an analytic manifold by a group of analytic homeomorphisms, PNAS 40 (9) 804-808 (1954) (doi:10.1073/pnas.40.9.804)

Since this traditional rational cohomology of orbifolds does, hence, not actually reflect the specific nature of orbifolds, a proposal for a finer notion of orbifold cohomology was famously introduced (motivated from orbifolds as target spaces in string theory, hence from orbifolding of 2d CFTs) in

• Weimin Chen, Yongbin Ruan, A New Cohomology Theory for Orbifold, Commun. Math. Phys. 248 (2004) 1-31 (arXiv:math/0004129)

However, Chen-Ruan cohomology of an orbifold $\mathcal{X}$ turns out to be just Borel cohomology with rational coefficients, hence is just Satake’s coarse cohomology – but applied to the inertia orbifold of $\mathcal{X}$. A review that makes this nicely explicit is (see p. 4 and 7):

• Emily Clader, Orbifolds and orbifold cohomology, 2014 (pdf)

Hence Chen-Ruan cohomology of a global quotient orbifold is equivalently the rational cohomology (real cohomology, complex cohomology) for the topological quotient space $AutMor(X\!\sslash\!G)/G$ of the space of automorphisms in the action groupoid by the $G$-conjugation action.

On the other hand, it was observed in (see p. 18)

• Ieke Moerdijk, Orbifolds as Groupoids: an Introduction, Alejandro Adem, Jack Morava, Yongbin Ruan (eds.) Orbifolds in Mathematics and Physics, Contemporary Math 310 , AMS (2002), 205–222 (arXiv:math.DG/0203100)

that for global quotient orbifolds Chen-Ruan cohomology indeed is equivalent to a $G$-equivariant Bredon cohomology of $X$ – for one specific choice of equivariant coefficient system (abelian sheaf on the orbit category of $G$), namely for $G/H \mapsto ClassFunctions(H)$.

Or rather, Moerdijk 02, p. 18 observes that the Chen-Ruan cohomology of a global quotient orbifold is equivalently the abelian sheaf cohomology of the naive quotient space $X/G$ with coefficients in the abelian sheaf whose stalk at $[x] \in X/G$ is the ring of class functions of the isotropy group at $x$; and then appeals to Theorem 5.5 in

• Hannu Honkasalo, Equivariant Alexander-Spanier cohomology for actions of compact Lie groups, Mathematica Scandinavica Vol. 67, No. 1 (1990), pp. 23-34 (jstor:24492569)

for the followup statement that the abelian sheaf cohomology of $X/G$ with coefficient sheaf $\underline{A}$ being “locally constant except for dependence on isotropy groups” is equivalently Bredon cohomology of $X$ with coefficients in $G/H \mapsto \underline{A}_x$ for $Isotr_x = H$. This identification of the coefficient systems is Prop. 6.5 b) in:

• Hannu Honkasalo, Equivariant Alexander-Spanier cohomology, Mathematica Scandinavia 63 (1988) 179-195 [doi:10.7146/math.scand.a-12232]

See also Section 4.3 of

• Richard Szabo, Alessandro Valentino, Ramond-Ramond Fields, Fractional Branes and Orbifold Differential K-Theory, Commun. Math. Phys. 294 (2010) 647-702 [doi:10.1007/s00220-009-0975-1, arXiv:0710.2773]

In summary:

• Traditional orbifold cohomology theory is Borel cohomology of underlying Borel construction-spaces, and reduces rationally further to the rational cohomology of underlying naive quotient spaces.

• Chen-Ruan cohomology is just the latter rational cohomology but applied after passage to the inertia orbifold. This is equivalent to the Bredon cohomology of the original orbifold, for one specific equivariant coefficient-system.

This suggests, of course, that more of proper equivariant cohomology should be brought to bear on a theory of orbifold cohomology. A partial way to achieve this is to prove for a given equivariant cohomology-theory that it descends from an invariant of topological G-spaces to one of the associated global quotient orbifolds.

For topological equivariant K-theory this is the case, by

• Dorette Pronk, Laura Scull, Prop. 4.1 in: Translation Groupoids and Orbifold Bredon Cohomology, Canad. J. Math. 62(2010), 614-645 (arXiv:0705.3249, doi:10.4153/CJM-2010-024-1)

Therefore it makes sense to define orbifold K-theory for orbifolds $\mathcal{X}$ which are equivalent to a global quotient orbifold $\mathcal{X} \simeq \prec(X \!\sslash\! G)$ to be the $G$-equivariant K-theory of $X$: $K^\bullet(\mathcal{X}) \;\coloneqq\; K_G^\bullet(X) \,.$

This is the approach taken in

• Alejandro Adem, Yongbin Ruan, Section 3 of: Twisted Orbifold K-Theory, Commun. Math. Phys. 237 (2003) 533-556 (arXiv:math/0107168)

Exposition and review of traditional orbifold cohomology, with an emphasis on Chen-Ruan cohomology and orbifold K-theory, is in:

• Alejandro Adem, Johann Leida, Yongbin Ruan, Orbifolds and Stringy Topology, Cambridge Tracts in Mathematics 171 (2007) (doi:10.1017/CBO9780511543081, pdf)

Proper orbifold cohomology

Discussion of orbifold cohomology in the context of Bredon cohomology:

• 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)

The suggestion to regard orbifold cohomology in global equivariant homotopy theory:

• Stefan Schwede, Introduction of: Orbispaces, orthogonal spaces, and the universal compact Lie group, Mathematische Zeitschrift 294 (2020), 71-107 (arXiv:1711.06019)

• Stefan Schwede, p. ix-x of: Global homotopy theory, New Mathematical Monographs, 34 Cambridge University Press, 2018 (arXiv:1802.09382)

Spelling out this suggestion of Schwede 17, Intro, Schwede 1, p. ix-x8:

• Branko Juran, Orbifolds, Orbispaces and Global Homotopy Theory (arXiv:2006.12374)

Based on results of or related to:

• André Henriques, David Gepner, Homotopy Theory of Orbispaces (arXiv:math/0701916)

The observation of the cohesion of global- over G-equivariant homotopy theory is due to:

• Charles Rezk, Global Homotopy Theory and Cohesion, 2014

A formulation of orbifold cohomology in the singular-cohesive $\infty$-topos of singular-smooth $\infty$-groupoids is then discussed in:

• Hisham Sati, Urs Schreiber, Proper Orbifold Cohomology (arXiv:2008.01101)

• Adrian Clough, Bastiaan Cnossen, Sil Linskens: Global spaces and the homotopy theory of stacks [arXiv:2407.06877]

On the resulting cohesive homotopy type theory with a pair of commuting cohesive structures:

• David Jaz Myers, Mitchell Riley, Commuting Cohesions [arXiv:2301.13780]