This may be regarded, as in [@F2], as the “path integral” over the fields on a point. Integration over $I{/\!/}G$, which is implemented as a categorical limit, amounts to taking $G$-invariants, which is a higher version of the Gauss Law in canonical quantization. It is also worth noting that the space of connections on $M={\ensuremath{S^1}}$ is the finite dimensional groupoid $G{/\!/}G$, where $G$ acts on $G$ by conjugation. Properly interpreted, this is the loop space of the fields on a point. On the other hand, for any manifold of dimension at least two, the stack of connections is infinite dimensional unless $G$ is finite.

In the latter case—for a finite group $G$—the path integral in all dimensions reduces to a finite sum, so is manifestly well-defined and satisfies the axioms of a field theory. This idea was initiated in [@F2], where the notion of a classical action in an $n$-dimensional field theory is extended to manifolds of all dimensions $<n$ and a higher categorical version of the path integral is used to heuristically define the quantum invariants.[^6] The developments in higher category theory in the intervening years, and the new definitions and structure theory for TQFT, make it possible to 