the finite element method in its most simple form considers piecewise polynomials on a triangulated space $\omega$. while the general theory does not require structure on $\omega$ beyond admitting a finite triangulation, it is quickly realized that in order to solve partial differential equations, $\omega$ must be endowed with a differentiable structure. we introduce a new framework which has the potential to allow the finite element method access to a broad class of differentiable manifolds of non-trivial topology.