Categorification of $\pi$?

number-theory category-theory groupoids

One approach to questions of this general nature is to find a groupoid whose groupoid cardinality is equal to the number in question; this is a form of groupoidification. For example, the groupoid cardinality of the groupoid $\text{FinSet}_0$ of finite sets and bijections is $\sum \frac{1}{n!} = e$, so this is a reasonable categorification of $e$; in fact arguably this is "the" categorification of $e$ and goes a long way towards explaining its prevalence in mathematics. Similarly, for any finite set $X$, the groupoid cardinality of the groupoid of $X$-colored finite sets and color-preserving bijections is $\sum \frac{|X|^n}{n!} = e^{|X|}$. Note that this groupoid is the coproduct of $|X|$ copies of $\text{FinSet}_0$; see also this math.SE question.

In a TWF188 John Baez mentions that he looked into the problem of finding a "natural" groupoid whose cardinality is $\pi$, but that he (and possibly some collaborators) weren't able to come up with any nice examples. So possibly this is the wrong direction to go in the particular case of $\pi$.

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