nLab equivariant group

Idea

For $G$ a fixed group, to be called the equivariance group, then by a $G$-equivariant group we mean a group object internal to $G$-actions, e.g. internal to G-sets, G-spaces, or G-manifolds, etc.

Beware that the term “equivariant group” for this notion is non-standard; or rather: there is no other established term for this notion at all. The term is meant to rhyme on the established terminology of equivariant principal bundles, in which context it serves to make nicely transparent the full (but often hidden) internal nature of the concept: equivariant principal bundles have, in general, equivariant groups as their structure groups.

Definition

We discuss equivariant groups in/as topological spaces, for definiteness and due to their relevance as models in equivariant homotopy theory. All of the discussion generalizes, say to smooth manifolds or general toposes (see at category of G-sets – For internal group actions).

Preliminaries

Definition

(convenient category of topological spaces)
We write

$$TopologicalSpaces \;\in\; Categories$$

for any convenient category of topological spaces whose mapping space serves as an internal hom, such as

• D-topological spaces

• compactly generated topological spaces.

This means in particular that for $X,Y,A \,\in\, TopologicalSpaces$, we have a natural bijection

(1)$$X \times Y \longrightarrow A \;\;\;\;\;\;\; \overset{adjuncts}{\leftrightarrow} \;\;\;\;\;\;\; X \longrightarrow Maps(Y,A)$$

between maps (meaning: continuous functions) out of the product topological space with $Y$ and maps into the mapping space.

Definition

(topological $G$-spaces)
For $G$ be a topological group – to be called the equivariance group – we write

$$G Actions(TopologicalSpaces) \;\in\; Categories$$

for the category whose

• objects$(X,\rho)$ are topological spaces $X$ equipped with continuous $G$-actions

  $$G \times X \overset{}{\longrightarrow} X \;\;\;\;\;\; \leftrightarrow \;\;\;\;\;\; G \overset{\rho}{\longrightarrow} Aut(X) \subset Maps(X,X) \,;$$

• morphisms are $G$-equivariant continuous functions between them (“maps”); i.e. for the category of “G-spaces”, often denote “$G Spaces$” or even $G Sp$ or similar.

Remark

(further conditions on the equivariance group)
For purposes of equivariant homotopy theory one typically assumes the topological equivariance group $G$ in Def. to be that underlying a compact Lie group, such as a finite group (as that guarantees that G-CW-complexes are well-behaved and that the equivariant Whitehead theorem holds). But for the plain point-set topology of equivariant groups and their equivariant bundles this condition is not necessary.

Remark

(topological G-spaces are cartesian monoidal) The category of topological G-spaces (Def. ) is a Cartesian monoidal category: The Cartesian product of two topological G-spaces $(X_i, \rho_i)$ is the underlying product topological space equipped with the diagonal action by $G$:

$$(X_1, \rho_1) \times (X_2, \rho_2) \;=\; \Big( X_1 \times X_2, \, \rho_{1,2}(g)(x_1,x_2) \,=\, \big( \rho_1(x_1), \rho_2(x_2) \big) \Big) \,.$$

In a Cartesian monoidal category there is a notion of internal group objects:

Equivariant groups

Definition

(equivariant topological groups)
Given an equivariance group $G$ (Def. ), a $G$-equivariant topological group $(\mathcal{G}, \alpha)$ is a group object internal to topological G-spaces (Def. ):

$$\big( \mathcal{G}, \, \alpha \big) \;\in\; Groups \big( G Actions ( TopologicalSpaces ) \big) \,.$$

(See Prop. below for the choice of notation used here.)

Remark

Since the forgetful functor from topological G-spaces (Def. ) to underlying topological spaces

$$\array{ G Actions ( TopologicalSpaces ) & \overset{ }{\longrightarrow} & TopologicalSpaces }$$

preservesCartesian products (explicitly so by Remark ), it preserves group objects and hence sends $G$-equivariant topological groups (Def. ) to underlying plain topological groups:

(2)$$\array{ Groups \big( G Actions ( TopologicalSpaces ) \big) & \overset{ }{\longrightarrow} & Groups ( TopologicalSpaces ) \\ \big( \mathcal{G} , \alpha \big) &\mapsto& \mathcal{G} \,. }$$

Properties

Equivalence with semidirect products with $G$

Proposition

(Equivariant groups as semidirect product groups)
The category of $G$-equivariant topological groups (Def. ) is equivalent

(3)$$\array{ Groups \big( G Actions ( TopologicalSpaces ) \big) & \overset{ \;\; \;\; }{\hookrightarrow} & Groups^{G/}_{/G} \\ \big( \mathcal{G}, \alpha \big) &\mapsto& \mathcal{G} \rtimes_{\alpha} G \,. }$$

to that of semidirect product groups of the form $(-) \rtimes G$

(4)$$\mathcal{G} \rtimes_\alpha G \;\; \coloneqq \;\; \Big( \mathcal{G} \times G \,, \;\;\; (\gamma_1, g_1) \cdot (\gamma_2, g_2) \;\coloneqq\; \big( \gamma_1 \cdot \alpha(g_1)(\gamma_2) ,\, g_1 \cdot g_2 \big) \Big)$$

and regarded as pointed objects in the slice category of Groups over $G$ via the canonical homomorphisms

(5)$$\array{ G &\longrightarrow& \mathcal{G} \times_\alpha G &\longrightarrow& G \\ && (\gamma,g) &\mapsto& g \\ g &\mapsto& (e_{\mathcal{G}},g) \,. }$$

(which jointly witness the semidirect product as a split group extension of $G$, see there).

Proof

This is a straightforward matter of unwinding the definitions:

First to see that we have a functor as claimed:

A group object in $G Actions(TopologicalSpaces)$ is, by definition, a plain topological group $\mathcal{G}$ whose underlying topological space is equipped with a continuous $G$-action and such such this $G$-action preserves all its group operations. In other words, this is a group $\mathcal{G}$ and a homomorphism $\alpha \;\colon\;G \longrightarrow Aut_{Grp}(\mathcal{G})$ to the group-automorphism group (whose hom-adjunct (1) is continuous).

This is exactly the data that determines the semidirect product group (4).

Moreover, a homomorphism of equivariant groups $(\mathcal{G}_1, \alpha_1) \longrightarrow (\mathcal{G}_2, \alpha_2)$ is a continuous group homomorphism $\phi \,\colon\, \mathcal{G}_1 \longrightarrow \mathcal{G}_2$ whose underlying map is $G$-equivariant in that

$$\underset{ g \in G }{\forall} \;\; \;\; \phi \circ \alpha_1(g) \;=\; \alpha_2(g) \circ \phi \,.$$

By (4) this means that $\phi$ induces a group homomorphism of semidirect product groups of the form

$$\array{ \mathcal{G}_1 \rtimes_{\alpha_1} G & \overset{ }{\longrightarrow} & \mathcal{G}_2 \rtimes_{\alpha_2} G \\ \mathcal{G}_1 \times G & \overset{ \phi \times id_G }{\longrightarrow} & \mathcal{G}_2 \times G }$$

This construction

(6)$$\array{ (\mathcal{G}_1, \alpha_1) && && \mathcal{G}_1 \rtimes_{\alpha_1} G \\ \big\downarrow {}^{_{\mathrlap{\phi}}} && \mapsto && \big\downarrow {}^{_{\mathrlap{\phi \times id_G}}} \\ (\mathcal{G}_2, \alpha_2) && && \mathcal{G}_2 \rtimes_{\alpha_2} G }$$

is clearly functorial.

It remains to see that this functor is a full subcategory-inclusion, hence a fully faithful functor, hence that it is a bijection on hom-sets for any pair of objects:

But by (6) the homomorphisms of semidirect product groups in its image are precisely those of the product form $\phi \times id_G$, and this is exactly the form of the homomorphisms between semidirect product groups that is picked out by slicing over and under $G$ (by Remark ).

Remark

Here and in the following we use that a group homomorphism out of a semidirect product group (4) is fixed already by its restriction to the two canonical subgroups

$$\array{ \mathcal{G} \\ \big\downarrow & \searrow^{ \mathrlap{ \gamma \mapsto \phi\big( \gamma, e_G \big) } } \\ \mathcal{G} \rtimes_{\alpha} G & \overset{\phi}{\longrightarrow} & \mathcal{G}' \rtimes_\alpha G \\ \big\uparrow & \nearrow_{ \mathrlap{ g \mapsto \phi\big(e_{\mathcal{G}},g\big) } } \\ G }$$

because every element of the semidirect product is equal to a product of elements from these subgroups:

(7)$$(\gamma,g) \;=\; (\gamma, e_G) \cdot ( e_{\mathcal{G}}, g ) \,.$$

This implies in particular that

1. $\phi$ being a morphism in $Groups_{/G}$ means equivalently that its restriction to $\mathcal{G}$ factors (via some group homomorphism $\mathcal{G} \to \mathcal{G}'$) through the canonical inclusion of $\mathcal{G}'$ (5);

2. $\phi$ being a morphism in $Groups^{G/}$ means equivalently that its restriction to $G$ is the canonical inclusion of $G$ (5).

Equivariant group actions

Proposition

(equivariant group actions as semidirect product group actions)
Under the identification from Prop. of $G$-equivariant groups $\big(\mathcal{G}, \alpha \big)$ with semidirect product groups $\mathcal{G} \rtimes_\alpha G$, we have an equivalence of their actions, given by:

(8)$$\array{ \big( \mathcal{G}, \, \alpha \big) Actions \big( G Actions ( TopologicalSpaces ) \big) & \overset{\simeq}{ \longrightarrow } & ( \mathcal{G} \rtimes G ) Actions( TopologicalSpaces ) \\ ( \mathcal{G}, \alpha ) \overset{ R }{ \to } Aut \big( (X, \rho) \big) &\mapsto& \mathcal{G} \rtimes_\alpha G \overset{ (R,\rho) }{\longrightarrow} Aut(X) }$$

with

(9)$$\underset{\gamma,g,x}{\forall} \;\;\; (R,\rho)(\gamma, g)(x) \;\coloneqq\; R(\gamma)\big( \rho(g)(x) \big) \,.$$

Proof

We observe that the given formula in fact establishes a bijection between the two kinds of actions:

First, notice by the decomposition (7) in Remark , that any action of $\mathcal{G} \rtimes_\alpha G$ can be written in the form (9) for some actions $R$ and $\rho$ of $\mathcal{G}$ and $G$, respectively, satisfying some compatibility conditions:

1. the action property of $\rho$ is equivalently the action property of $(R,\rho)$ on elements of the form $(e, g)$;

2. the action property of $R$ on the underlying topological spaces is equivalently the action property of $(R,\rho)$ on elements of the form $(\gamma,e)$;

3. the $G$-equivariance of $R$

   (10)$$\underset{ \gamma,g,x }{\forall} \;\;\; R \big( \alpha(g)(\gamma) \big) \big( \rho(g)(x) \big) \;=\; \rho(g) \big( R(\gamma)(x) \big)$$

   is equivalent to the action property of $(R,\phi)$ on mixed pairs of elements of the form $\big( (e_{\mathcal{G}},g), \; (\gamma,e_G) \big)$:

   $$\underset{ \gamma,g,x }{\forall} \;\;\; (R,\rho) \big( \underset{ \big( \alpha(g)(\gamma), g \big) }{ \underbrace{ (e,g) \cdot (\gamma,e) } } \big) (x) \;=\; (R,\rho)(e,g) \big( (R,\rho)(\gamma,e) (x) \big) \,.$$

These three conditions exhaust the conditions on $R$ to be a $G$-equivariant action. Therefore it just remains to see that they also exhaust the conditions on $(R,\rho)$ to be a plain action:

But the remaining mixed action conditions on $(R, \phi)$

$$\underset{ \gamma_1, \gamma_2, g, x }{\forall} \;\;\; (R,\rho) \big( \underset{ = \, (\gamma_1 \cdot \gamma_2,g) }{ \underbrace{ (\gamma_1,e) \cdot (\gamma_2,g) } } \big) (x) \;=\; (R,\rho) \big( (\gamma_1,e) \big) \Big( (R,\rho) \big( (\gamma_2,g) \big) (x) \Big)$$

and

$$\begin{aligned} \underset{ \gamma, g_1, g_2, x }{\forall} \;\;\; (R,\rho) \big( (e,g_1) \cdot (\gamma, g_2) \big) (x) & =\; (R,\rho) \Big( \big( \alpha(g_1)(\gamma), g_1 \cdot g_2 \big) \Big) (x) \\ & =\; R \big( \alpha(g_1)(\gamma) \big) \big( \rho(g_1 \cdot g_2) (x) \big) \\ & \overset{!}{=} \; R \big( \alpha(g_1)(\gamma) \big) \Big( \rho(g_1) \big( \rho(g_2) (x) \big) \Big) \\ & \overset{!}{=} \; \rho(g_1) \Big( R \big( \gamma \big) \big( \rho(g_2) (x) \big) \Big) \\ & =\; (R,\rho) \big( (e,g_1) \big) \Big( (R,\rho) \big( (\gamma, g_2) \big) (x) \Big) \end{aligned}$$

follow right from the definition (9) and using again (in the highlighted steps “$\overset{!}{=}$”):

• the action propery of $\rho$ from item 2 above;

• the $G$-equivariance of $R$ (10) from item 3 above.

In conclusion, $(R,\rho)$ is an action of the semidirect product group $\mathcal{G} \rtimes_\alpha G$ on $X$ iff $R$ is a $G$-equivariant action of $\mathcal{G}$ on $X$.

This implies immediately that the condition for a map $X \to X$ to be an action homomorphisms on both sides are the same.

And so the functor (8) is in fact an isomorphism on both objects as well as morphisms, hence in particular is an equivalence of categories.

Related concepts

• equivariant bundle

• equivariance group, structure group, gauge group