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G-set (changes) in nLab

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Context

Group Theory

group theory

Classical groups

Finite groups

Group schemes

Topological groups

Lie groups

Super-Lie groups

Higher groups

Cohomology and Extensions

Related concepts

Representation theory

representation theory

geometric representation theory

Ingredients

representation, 2-representation, ∞-representation

Geometric representation theory

Contents

Idea

Given a topological group GG, a (continuous) GG-set is a set XX equipped with a continuous group action μ:G×X→X\mu: G \times X \to X, where XX is given the discrete topology.

In the case where GG is a discrete group, the continuity requirement is void, and this is just a permutation representation of the discrete group GG.

Note that since XX must be given the discrete topology, this behaves rather unlike topological G-spaces. In particular, a topological group does not act continuously on itself, in general. Thus this notion is not too useful when GG is a “usual” topology group like SU(2)SU(2). Instead, the topology on the group acts as a filter of subgroups (where the filter contains the open subgroups), and each element of a continuous GG-set is required to have a “large” stabilizer.

The GG-sets form a category, where the morphisms are the GG-invariant maps. See category of G sets.

Properties

Relation to GG-orbits

\begin{remark}\label{GActionsAndGOrbits} ($G$-sets are the free coproduct completion of $G$-orbits) \linebreak Let G∈Grp(Set)G \,\in\, Grp(Set) be a discrete group. Since every G-set XX decomposes as a disjoint union of transitive actions, namely of orbits of elements of XX, the defining inclusion of the orbit category into $G Set$ exhibits the latter as the free coproduct completion of the orbit category (see also this Prop.). \end{remark}

For topological groups

Proposition

Let GG be a topological group, and XX be a set with a GG action μ:G×X→X\mu: G \times X \to X. Then the action is continuous if and only if the stabilizer of each element is open.

Proof

Suppose μ\mu is continuous. Since XX has the discrete topology, {x}\{x\} is an open subset of XX. So μ −1({x})\mu^{-1}(\{x\}) is open. So we know the stabilizer

I x={g∈G:g⋅x=x}={g∈G:(g,x)∈μ −1({x})} I_x = \{g \in G: g \cdot x = x\} = \{g \in G: (g, x) \in \mu^{-1}(\{x\})\}

is open.

Conversely, suppose each such set is open. Given any (necessarily open) subset A⊆XA \subseteq X, its inverse image is

μ −1(A)=⋃ a∈Aμ −1({a}). \mu^{-1}(A) = \bigcup_{a \in A} \mu^{-1}(\{a\}).

So it suffices to show that each μ −1({a})\mu^{-1}(\{a\}) is open. We have

μ −1({a})=⋃ x∈X{g∈G:g⋅x=a}×{x}. \mu^{-1}(\{a\}) = \bigcup_{x \in X}\{g \in G: g \cdot x = a\} \times \{x\}.

Thus we only have to show that for each a,x∈Xa, x \in X, the set {g∈G:g⋅x=a}\{g \in G: g \cdot x = a\} is open. If there is no such gg, then this is empty, hence open. Otherwise, let g 0g_0 be such that g 0⋅x=ag_0 \cdot x= a. Then we have

{g∈G:g⋅x=a}=g 0⋅I x. \{g \in G: g \cdot x = a\} = g_0 \cdot I_x.

Since g 0g_0 is a homeomorphism, and I xI_x is open, this is open. So done.

Examples

Discrete groups

In the following examples, all groups are discrete.

Continuous groups

  • The group Σ N\Sigma_N of permutations of the natural numbers can be given the topology generated by the stabilizers of finite subsets of NN. This acts continuously on NN. This is used in the construction of the basic Fraenkel model. See also nominal sets.

References

An early account (where the term “representation group” is used to refer to a finite set equipped with a permutation action):

  • William Burnside, On the Representation of a Group of Finite Order as a Permutation Group, and on the Composition of Permutation Groups, Proceedings of the American Mathematical Society 1901 (doi:10.1112/plms/s1-34.1.159)

For a more modern account see

Basic exposition of GG-sets as a Grothendieck topos:

  • Jaap van Oosten, (eg. p. 32 in:) Topos Theory (2018) [[pdf](https://webspace.science.uu.nl/~ooste110/syllabi/topostheory.pdf), pdf]

Last revised on November 29, 2023 at 16:30:36. See the history of this page for a list of all contributions to it.