  We formalize Feynman's construction of the quantum mechanical path integral.
To do this, we shift the emphasis in differential geometry from the tangent
bundle onto the pair groupoid. This allows us to use the van Est map and the
piecewise linear structure of manifolds to develop a coordinate-free, partition
of unity-free approach to integration of differential forms, etc. This
framework makes sense for any field theory valued in a Lie algebroid. We apply
it to define the Wiener measure, stochastic integrals and other observables in
a coordinate-free way. We use it to reconstruct Chern-Simons with finite gauge
group and to obtain some non-perturbative deformation quantizations via the
Poisson sigma model on a disk.
