nLab permutation groupoid

Definition

The (skeleton of the) core groupoid of FinSet, hence the category of finite sets and bijections (permutations) between them, is sometimes called the permutation category or the permutation groupoid (e.g. BMT 2021).

It deserves to be called the symmetric groupoid, because its connected components are the delooping groupoids of all the symmetric groups $Sym(n)$:

$$Core(FinSet) \;\simeq\; \textstyle{ \underset{n \in \mathbb{N}} {\coprod} } \, \mathbf{B} Sym(n) \,.$$

In its skeletal incarnation on their right , this carries the structure of a strict symmetric monoidal category with addition of natural numbers as its tensor product. As such it is the free strict symmetric monoidal category on one object (namely on $1 \in \mathbb{N}$).

The presheaves on the permutation groupoid are also known as combinatorial species.

Related concept

• FinSet

• symmetric group

• combinatorial species

• braid category

References

• Wikipedia, Permutation category

• John Baez, Joe Moeller, Todd Trimble, Schur functors and categorified plethysm [arxiv:2106.00190]