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:
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
following analogous constructions in
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
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):
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)
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
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:
See also Section 4.3 of
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
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
Exposition and review of traditional orbifold cohomology, with an emphasis on Chen-Ruan cohomology and orbifold K-theory, is in:
Last revised on March 14, 2021 at 12:15:18. See the history of this page for a list of all contributions to it.