Dear John, This result is also proved in my paper below, published in 2001. M. Grandis, Finite sets and symmetric simplicial sets, Theory Appl. Categ. 8 (2001), No. 8, 244-252. Abstract. The category of finite cardinals (or equivalently, of finite sets) is the symmetric analogue of the category of finite ordinals, and the ground category of a relevant category of presheaves, the augmented symmetric simplicial sets. We prove here that this ground category has characterisations similar to the classical ones for the category of finite ordinals, by the existence of a universal symmetric monoid, or by generators and relations. The latter provides a definition of symmetric simplicial sets by faces, degeneracies and transpositions, under suitable relations. Best wishes, Marco On 05/gen/2016, at 00.13, John Baez wrote:
Dear Categorists -
A student of mine is wondering who first noticed this fact: if you take a skeleton of the category of finite sets and make it into a strict symmetric monoidal category using cartesian product, it's the "free strict symmetric monoidal category on a commutative monoid object". Or in other words, it's the PROP for commutative monoids.
He noticed that in 2001, Teimuraz Pirashvili wrote a paper "On the PROP corresponding to bialgebras":
http://arxiv.org/abs/math/0110014
Pirashvili says this fact is "well known", and gives a proof, but no reference.
Can you help us dig deeper? It's just a matter of getting the history right.
Best, jb
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