Yesterday I described the answer to the puzzle of what the generating function

counts by sketching a proof of the following more general identity: if is a finitely generated group and is the number of subgroups of of index , then

.

The main ingredient is the exponential formula, but the discussion of the proof involved some careful juggling to make sure we weren’t inappropriately quotienting out by various symmetries, and one might find this conceptually unsatisfying. The goal of today’s post is to state a categorical result which describes exactly how to juggle these symmetries and gives a conceptually clean proof of the above identity.

The key is to describe in exactly what sense a finite -set (corresponding to the LHS) has a canonical “connected components” decomposition as a disjoint union of transitive -sets (corresponding to the RHS), which is the following.

**Claim:** The symmetric monoidal groupoid of finite -sets, with symmetric monoidal structure given by disjoint union, is the free symmetric monoidal groupoid on the groupoid of transitive finite -sets.

From here, we’ll use a version of the exponential formula that comes from relating (weighted) groupoid cardinalities of a groupoid and of the free symmetric monoidal groupoid on it.