# nLab 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.

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

Revised on July 9, 2010 01:22:17 by Toby Bartels (173.60.119.197)