# Contents

## Definition

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.

###### Warning

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

## Examples

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)