A Pi Groupoid

Recall that the cardinality of a groupoid involves the inverse of the cardinalities of groups. At PI, Jeff Morton told me about a very nice example involving, for instance, the cyclic groups $C_{n} \times C_{n}$, which each have cardinality $n^2$. That is, we can have a cardinality $\pi^2$, because

$\pi^{2} = 6 \sum_{k} \frac{1}{k^2}$.

Recall that this infinite sum is the number $\zeta (2)$ for the Riemann zeta function, first evaluated by Euler in 1735. Since $e$ is also a groupoid cardinality, namely for the groupoid of finite sets and bijections, it seems that transcendentals naturally appear in the context of infinite groupoids.