down to points which is manifestly local and functorial. We write these finite sums explicitly in this section. In §\[sec:5\] we indicate one possible path which should put these ideas on a rigorous footing.

For the 2-dimensional finite gauge theory based on the central extension ${\mathbb T}\to{G^\tau }\to G$ the classical invariant attached to a $G$-bundle $P\to X$ over a closed oriented 2-manifold is $$I(X,P) = e^{2\pi i \sigma _X(P)},$$ where $\sigma \in H^2(BG;{{\mathbb Q}}/{{\mathbb Z}})\cong H^3(BG;{{\mathbb Z}})$ is the characteristic class corresponding to the central extension (see Proposition \[thm:6\]) and $\sigma _X(P)\in {{\mathbb Q}}/{{\mathbb Z}}$ is the characteristic number of the bundle. Physicists call $I(X,P)$ the ‘exponentiated action’ of the field $P$ on $X$. The quantum invariant of $X$ is then $$\label{eq:31}
     F(X)= \sum\limits_{P}\frac{1}{\#\operatorname{Aut}P}\;I(X,P),$$ where the sum is over a set of representative $G$-bundles on $X$, one in each equivalence class. For the trivial central extension $I(X,P)=1$ for all $P$ and  counts the number of representations of $\pi _1X$ into the finite group $G$. (There is an overall factor of $1/\#G$.) In that case we can extend to a theory of unoriented manifolds.

Now consider this 2-dimensional theory over the circle ${\ensuremath{S^1}}$. Recall from the introduction that the groupoid 