Why is $ \mathbb{N} $ a fine moduli space for the moduli problem of finite sets up to bijection?

Question

How to establish rigourously that $ \mathbb{N} $ is a fine moduli space for the moduli problem of finite sets up to bijection ?.

Thanks in advance for your help.

Answer 1

@AreaMan already wrote the answer in the comments but let's be explicit.

Let $U = \{(n,m) \in \Bbb N^2 : 1 \leq m \leq n \}$. I claim that $U$ is the universal family with $p : U \to \Bbb N$ being the projection onto the first coordinate.

Now let $f : X \to S$ a family of finite sets, i.e $f^{-1}(s)$ is finite for all $s \in S$. I can consider the map $g : S \to \Bbb N, s \mapsto |f^{-1}(s)|$. It is clear that up to a fiberwise bijection the pullback of $U$ by $g$ is $X$, as you can easily check from the definition. This is just a fancy way of saying that if we look at the set $ g^*U := \{ (s,u) \in S \times U : g(s) = p(u)\}$ then we have a natural bijection $g^*U \cong X$.

So every family $X \to S$ is naturally equivalent to a morphism $g : S \to \Bbb N$, i.e we have a natural isomorphism $F(S) \cong \text{Hom}_{\text{Set}}(S, \Bbb N)$, and this exactly means that $\Bbb N$ is a fine moduli space for the moduli problem of finite sets up to bijection.