TITLE: What is the "right" definition of the homology(cohomology) of an orbifold?
QUESTION [11 upvotes]: What is the "right" analog in the orbifold case of a singular homology of a topological space?
We can not just take the homology of the underlying space, because it does not contain much information.
For example, is there any kind homology of orbifolds such that the first homology group is the abelianization of the fundamental group of the orbifold? And such that an $n$-dim orbifold will have all homology group higher than $n$-dim equal to zero? And if there is such a homology, will there be Poincare duality in the orbifold case???
Thanks very much!

REPLY [8 votes]: If your orbifold is given to you as a proper etale Lie/topological groupoid, you can:

take the nerve of this groupoid, to get a simplicial manifold/space, 
take the levelwise singular set to get a bisimplicial set, 
take the diagonal to get a simplicial set,  
take geometric realisation to get a topological space, 
and then take (co)homology. 

This recipe at the very least gives you the homotopy type of the orbifold as a space (to which you can apply the (co)homology functor you want), but it perhaps too big and unwieldy for your purposes.