permutation representation

Let $C$ be a groupoid.

A *permutation representation* of $C$ is a representation of $C$ on Set, i.e. a functor $C \to \Set$.

A *linear permutation representation* is a functor $C \to$ Vect that factors through a permutation representation via the free functor $k^{|-|}\colon Set \to Vect$ which sends a set to the vector space for which this set is a basis.

In the usual literature of representation theory, “linear permutation representations” are just called “permutation representations”.

Notably for $C = \mathbf{B}G$ the delooping groupoid of a group $G$, a permutation representation $\mathbf{B}G \to Set$ is a set equipped with a $G$-action.

The category

$Rep(G, Set) \simeq PSh(\mathbf{B} G)$

is the classifying topos for the group $G$.

For other general perspectives on this see also at *infinity-action* the section *Examples – Discrete group actions on sets*.

Revised on October 15, 2015 12:55:10
by Todd Trimble
(67.81.95.215)