In the paper "A new cohomology theory for orbifold" by Chen/Ruan, they define the *orbifold cohomology group* of an orbifold $X$ by


$H^d\_{orb}(X)=\bigoplus\_{(g) \in T} H^{d-2\iota\_{(g)}}(X\_{(g)})$


where $H^\ast(X\_{(g)})$ is the singular cohomology of $X\_{(g)}$ with real coefficients and $\iota(g)$ is a rational number. 


Also, the *orbifold Dolbeault cohomology group* is defined by


$H^{p,q}\_{orb}(X)=\bigoplus\_{(g) \in T}H^{p-\iota\_{(g)},q-\iota\_{(g)}}(X\_{(g)})$.


How are these cohomologies with rational coefficients defined?


EDIT: In Adem/Leida/Ruan's "Orbifolds and stringy topology" they similarly define the Chen-Ruan cohomology group for orbifold groupoid as


$H^d\_{CR}=\bigoplus\_{(g) \in T}H^d(X\_{(g)})[-2\iota\_{(g)}]=\bigoplus\_{(g) \in T}H^{d-2\iota\_{(g)}}(X\_{(g)})$,


but I am also unfamiliar with the notation $H^d(X\_{(g)})[-2\iota\_{(g)}]$.



Rationally graded singular cohomology
It is my understanding that $H^d$ is taken to be trivial unless $d$ is an integer. Hence this is just notation for a degree shift by a rational number.


Similarly, in the other formula, the square bracket notation is also just a degree-shift.


