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Let $F$ be a combinatorial species. The exponential generating series of $F$ is defined to be $$ \sum_{n \geq 0} \frac{ \lvert F_n \lvert x^n}{n!} $$ It was observed by Baez and Dolan in their paper "From finite sets to Feymann diagrams" that the exponential generating series can be categorified to a 1-groupoid $$ \sum_{n \geq 0} F_n \times X^n // S_n $$ You recover the exponential generating series by taking groupoid cardinality and setting $ x = \lvert X \lvert $. The cycle index series is another important series attached to a combinatorial species. It is defined by $$ Z_F(x_1,x_2,\dots) = \sum_{n \geq 0} \frac{1}{n!} \sum_{\sigma \in S_n} \lvert {\rm Fix}(F[\sigma])\lvert x_1^{\sigma_1} x_2^{\sigma_2} \dots $$

Question: Is it possible to construct an $n$-groupoid whose groupoid cardinality is $Z_F(x_1,x_2,\dots)$?

By $n$-groupoid, I mean a homotopy $n$-type?

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  • $\begingroup$ Why $n$-groupoids? I don't see any need to go past groupoids here. $\endgroup$
    – Qiaochu Yuan
    Mar 2, 2015 at 4:55
  • $\begingroup$ I have no reason to think that n > 1 except that I couldn't write down a groupoid which worked. Obviously that doesn't mean it is not possible though! $\endgroup$
    – Daniel Barter
    Mar 2, 2015 at 6:41
  • $\begingroup$ It's straightforward to write down a groupoid that works, the question is whether it can be described in a nice way: it's the groupoid of tuples of a permutation $\sigma \in S_n$, a fixed point $f \in F_n$, and functions $c_i(\sigma) \to X_i$ where $c_i$ denotes the set of $i$-cycles and $X_i$ are some auxiliary groupoids. There is an $S_n$-action on this data by $\tau : (\sigma, f) \mapsto (\tau \sigma \tau^{-1}, \tau f)$ (and identifying $i$-cycles of $\sigma$ with $i$-cycles of its conjugate) and that's where the $\frac{1}{n!}$ comes from. $\endgroup$
    – Qiaochu Yuan
    Mar 2, 2015 at 18:00

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