nLab
permutation groupoid

The permutation groupoid, sometimes denoted $\mathbb{P}$, is a skeleton of the groupoid of finite sets and bijections. Namely:

$$\mathbb{P}=\underset{n\ge 0}{⨆}{S}_n,$$\mathbb{P} = \bigsqcup_{n \ge 0} S_n \, ,

where objects are natural numbers, all morphisms are automorphisms, and the automorphism group of the object $n$ is the symmetric group $S_n$.

In other words, $\mathbb{P}$ is equivalent to the core of FinSet.

There are many notations for $\mathbb{P}$ besides ‘$\mathbb{P}$’, such as $S$ and $\Sigma$. In The Joy of Cats, $\mathbb{P}$ is denoted $\mathrm{Bij}$.