to emphasize the two tiers.

Let $n=1$. Then a theory $F$ assigns a vector space $F(\operatorname{pt}_+)$ to the positively oriented point. We use the oriented interval with all possibilities for decomposing the boundary into incoming and outgoing components to deduce that $F(\operatorname{pt}_-)$ is the dual space of $F(\operatorname{pt}_+)$ and the intervals give the duality pairing. Finally, $F({\ensuremath{S^1}})=\dim F(\operatorname{pt})$, and the entire theory is determined up to isomorphism by this nonnegative integer.

We now describe 1-dimensional pure gauge theory with compact gauge group $G$ and action given by a class in $H^2(BG;{\mathbb Z})$. Let ${\mathbb T}\subset {{\mathbb C}}$ be the circle group.

\[thm:1\] There is an canonical isomorphism $\operatorname{Hom}(G,{\mathbb T})\xrightarrow{\cong }H^2(BG; {\mathbb Z})$.

In purely topological terms the classifying space $BG$ carries a universal principal $G$-bundle, and the isomorphism assigns to an abelian character $G\to{\mathbb T}$ the Chern class of the associated principal ${\mathbb T}$-bundle. A more rigid viewpoint: abelian characters are in 1:1 correspondence with isomorphism classes of principal ${\mathbb T}$-bundle on the groupoid $*{/\!/}G$. A proof that the latter are classified by $H^2(BG; {\mathbb Z})$ may be found in several texts, e.g. [@AS Proposition 6.3], [@Bry Appendix].

Let $\lambda {\colon}G\to{\mathbb T}$ be an abelian character. Then the *classical* 1-dimensional 