  We relate Kellendonk and Putnam's pattern-equivariant (PE) cohomology to the
inverse-limit structure of a tiling space. This gives a version of PE
cohomology with integer coefficients, or with coefficients in any Abelian
group. It also provides an easy proof of Kellendonk and Putnam's original
theorem relating PE cohomology to the Cech cohomology of the tiling space. The
inverse-limit structure also allows for the construction of a new non-Abelian
invariant, the PE representation variety.
