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 $\zeta \left(2\right)$ 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.