permutation groupoid

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

$\mathbb{P} = \bigsqcup_{n \ge 0} S_n \, ,$

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.

$\mathbb{P}$ can be made into a strict symmetric monoidal category with addition as its tensor product, and it is then the free strict symmetric monoidal category on one object (namely $1$).

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