nLab
permutation representation

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 $k^{\mid -\mid }:\mathrm{Set}\to \mathrm{Vect}$ which sends a set to the vector space for which this set is a basis.

Warning

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

Examples

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

The category

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

is the classifying topos for the group $G$.

Related entries

• representation theory, linear representation, category of representations

Related concepts

• ∞-permutation representation