permutation groupoid

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

= n0S 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 nn is the symmetric group S nS_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 11).

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

Revised on September 6, 2017 03:57:50 by John Baez (