Pattern-Equivariant Cohomology with Integer Coefficients

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.