  We introduce a version of the P=W conjecture relating the Borel-Moore
homology of the stack of representations of the fundamental group of a genus g
Riemann surface with the Borel-Moore homology of the stack of degree zero
semistable Higgs bundles on a smooth projective complex curve of genus $g$. In
order to state the conjecture we propose a construction of a canonical
isomorphism between these Borel-Moore homology groups. We relate the stacky P=W
conjecture with the original P=W conjecture concerning the cohomology of smooth
moduli spaces of twisted objects, and the PI=WI conjecture concerning the
intersection cohomology groups of singular moduli spaces of untwisted objects.
In genus zero and one, we prove the conjectures that we introduce in this
paper.
