permutation representation



Let CC be a groupoid.

A permutation representation of CC is a representation of CC on Set, i.e. a functor CSetC \to \Set.

A linear permutation representation is a functor CC \to Vect that factors through a permutation representation via the free functor k ||:SetVectk^{|-|}\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=BGC = \mathbf{B}G the delooping groupoid of a group GG, a permutation representation BGSet\mathbf{B}G \to Set is a set equipped with a GG-action.

The category

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

is the classifying topos for the group GG.

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

Revised on January 24, 2017 07:01:04 by Urs Schreiber (