nLab category of G-sets

Idea

This entry is about the category of continuous G-sets, for a topological group $G$. Continuous $G$-sets are sets $X$ with an action of $G$ that is continuous when $X$ is given the discrete topology.

The category of continuous $G$-sets is a Grothendieck topos, and is closely related to Fraenkel-Mostowski models.

Definition

Let $G$ be a topological group.

Definition

The category of continuous $G$-sets, to be denoted $G Set$, is the category of sets $X$ equipped with a continuous (left) $G$-action $\mu \colon G \times X \to X$, where $X$ is given the discrete topology, and the morphisms are the $G$-equivariant functions.

There is a simple characterization of when a $G$-action is continuous.

Proposition

Let $G$ be a topological group, and $X$ be a set equipped with a $G$ action $\mu \colon G \times X \to X$. Then the action is continuous if and only if the stabilizer subgroup of each element is open.

Proof

See G-sets.

Properties

Notation

For a topological group $G$, we write $G^\delta$ for its underlying discrete group. Every $G$-set can be viewed as a $G^\delta$ set in the obvious way.

Proposition

The inclusion $i_G \colon G Set \to G^\delta Set$ has a right adjoint $r_G: G^\delta \to G Set$ such that for $X \in G Set$, the continuous $G$-set $r_G X$ is the subset of $X$ consisting of those points with open stabilizer subgroup.

Proof

This follows from the observation that if $f \colon X \to Y$ is a $G$-equivariant function between $G$-sets, then for each $x \in X$, the stabilizer of $f(x)$ includes the stabilizer of $x$.

Proposition

The forgetful functor $U \colon G^\delta Set \to Set$ creates all small limits and colimits.

Proof

This follows from the general fact that limits and colimits in presheaf categories are computed pointwise.

Alternatively, there is an obvious $G$-action we can put on the limits or colimits of the underlying sets of the $G$-sets.

The creating limits part also comes from the fact that the forgetful functor is monadic.

Proposition

The inclusion $i_G: G Set \to G^\delta Set$ creates all finite limits and all colimits.

Proof

This follows from the observation that the finite limits and colimits created by $U: G^\delta Set \to Set$ have a continuous action if each factor has a continuous action, using the fact that finite intersections and arbitrary unions of open sets are open.

Corollary

The category $G Set$ has all finite limits and arbitrary colimits.

Corollary

The adjunction $i_G \dashv r_G$ is a geometric morphism.

Corollary

A map in $G Set$ is a monomorphism if and only if it is injective; epimorphism if and only if it is surjective.

Proof

A map is monic if and only if its kernel pair is an isomorphism, and similarly for epic, and the forgetful functor to $Set$ preserves all finite limits and colimits.

Corollary

The subobject classifier of $G Set$ is $\Omega = 2$, the two-point set with the trivial $G$-action.

Proposition

The exponential object in $G^\delta Set$ is defined by $Y^X = \{\text{functions}\;X \to Y\}$, with the $G^\delta$-action given by

$$g \cdot f = g \circ f \circ g^{-1}.$$

The exponential object $Y^X$ in $G Set$ is given by the subset of the functions $X \to Y$ that have an open stabilizer, ie.

$$Y^X = r_G(i_G(Y)^{i_G(X)}).$$

Proof

Let $A, X, Y$ be $G^\delta$-sets. Given a map $f:A \times X \to Y$, we obtain $\hat{f}:A \to Y^X$ by

$$\hat{f}(a)(x) = f(a, x).$$

We now check that $\hat{f}$ is $G^\delta$-invariant — we have

$$(g \cdot \hat{f}(a))(x) = g \cdot \hat{f}(a)(g^{-1}\cdot x) = g \cdot f(a, g^{-1}x) = f(g \cdot a, x) = \hat{f}(g \cdot a)(x).$$

So we get

$$g \cdot \hat{f}(a) = \hat{f}(g \cdot a).$$

Conversely, given a $\hat{f}: A \to Y^X$, we obtain $f: A \times X \to Y$ by $f(a, x) = \hat{f}(a)(x)$, and we have

$$g \cdot f(a, x) = g \cdot \hat{f}(a)(x) = (g \cdot \hat{f}(a))(g\cdot x) = \hat{f}(g \cdot a)(g \cdot x) = f(g \cdot a, g \cdot x).$$

So $f$ is $G^\delta$-invariant. It is clear that these operations are inverses to each other, and straightforward computations show that this is natural in $A$, $X$ and $Y$.

Then in general, if $A, X, Y$ are in fact $G$-sets, then we can compute

$$\begin{aligned} G Set\left(A, r_G(i_G(Y)^{i_G(X)})\right) &= G^\delta Set \left(i_G(A), i_G(Y)^{i_G(X)}\right)\\ &= G^\delta Set\left(i_G(A) \times i_G(X), i_G(Y)\right)\\ &= G^\delta Set\left(i_G(A \times X), i_G(Y)\right)\\ &= G Set\left(A \times X, Y\right), \end{aligned}$$

using the fact that $r_G(i_G(X)) = X$ for all $X \in G Set$.

Corollary

The power object of $X \in G Set$ is given by the subsets of $X$ that have an open stabilizer.

In conclusion we have:

Theorem

The category $G Set$ is an elementary topos.

Equivalent characterizations

As a Grothendieck topos

Theorem

Let $G$ be a topological group. Then the category $G Set$ is equivalent to the topos of sheaves on the atomic site $(S(G), At)$, where the objects of $S(G)$ are the open subgroups of $G$, and the morphisms $H \to K$ are the left cosets $a K$ such that $H \subseteq a K a^{-1}$, and all non-empty sieves are covering.

Alternatively, it is the full subcategory of $G Set$ containing objects of the form $G/U$, where $U$ is an open subgroup.

Proof

See MacLane and Moerdijk, Chapter III.9.

More generally, by the comparison lemma, we have

Theorem

Let $G$ be a topological group, and $\mathcal{U}$ be a cofinal set of open subgroups, ie. every open subgroup contains a member of $\mathcal{U}$. Then the category $G Set$ is equivalent to the topos of sheaves on the atomic site $(S(G, \mathcal{U}), At)$, where the objects of $S(G, \mathcal{U})$ are the open subgroups in $\mathcal{U}$, and the morphisms $H \to K$ are the left cosets $a K$ such that $H \subseteq a K a^{-1}$, and all non-empty sieves are covering.

Alternatively, it is the full subcategory of $G Set$ containing objects of the form $G/U$, where $U \in \mathcal{U}$.

In particular, if $G$ is a discrete group, then the trivial subgroup itself is a cofinal set of open subgroups. So $G Set$ is the category of sheaves on the category with only one object, whose morphisms are the elements of $G$. This is the usual characterization of $G Set$ as the functor category $Set^G$.

As a comonad algebra

To be included.

As a classifying topos

To be included.

For internal group actions

The above construction of the category $G Set$ – of G-sets for a discrete group $G$ in the topos Set of sets – generalizes to group objects $G \in Groups(\mathcal{E})$ in any topos $\mathcal{E}$:

The resulting category $G\mathcal{E}$ – of objects of $\mathcal{E}$ equipped with internal $G$-action and with $G$-equivariant morphisms between them – is itself a topos, by the same proof as above Thm .

A particularly interesting case of this is an internal group $H$ in the topos $G Set$ itself, for $G$ a discrete group (a “$G$-equivariant group”, see there for more):

Seen externally this is equivalently a discrete group $H$ equipped with a group homomorphism $G \to \Aut(H)$ to the automorphism group of $H$. Notice that this data induces the corresponding semidirect product $H \rtimes G$. With that, we have:

Proposition

Let $G$ be a discrete group, and let $H$ be a group object in $G Set$. Then the category $H (G Set)$ is equivalent to $(H \rtimes G) Set$.

Proof sketch

This is a straightforward computation. Given an $X \in H (G Set)$, we write $\mu \colon H \times X \to X$ for the action morphism, and denote the action of $G$ merely by a dot. Then we define the action of $H \rtimes G$ by

$$((h, g), x) \mapsto \mu(h, g \cdot x) \,.$$

Conversely, given an object $X$ with an $H \rtimes G$ action, we give it a $G$ action by $g \cdot x = (e, g) \cdot x$, and an $H$-action by $\mu(h, x) = (h, e) \cdot x$.

The case of topological groups is more complicated, because an internal topology $T$ on a space $X$ is an internally complete lattice $T \subseteq P(X)$, which is not necessarily closed under infinite external unions. However if we do the rather unnatural (?) thing of closing it under all external unions, then we make $X$ an external topological space. Then we have the following result:

Proposition

Let $G$ be a topological group, and let $H$ be a topological group object in $G Set$. Then $H (G Set)$ is equivalent to $(H \rtimes G) Set$, where $H \rtimes G$ is given the product topology.

The proof is a straightforward check that the continuity conditions match up.

Related concepts

• G-set

• topological G-space

• permutation representation

• Fraenkel-Mostowski model

• class equation

References

Basic exposition of the topos of $G$-sets for discrete groups:

• Jaap van Oosten, (eg. p. 32 in:) Topos Theory (2018) [pdf, pdf]

Some elementary properties of continuous $G$-sets can be found in books such as

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic, Springer (1992) [doi:10.1007/978-1-4612-0927-0]

The formal correspondence between permutation models of ZFA and toposes of continuous $G$-sets can be found in

• Michael Fourman, Sheaf models for set theory, Journal of Pure and Applied Algebra, 19 (1980) pp 91-101, doi:10.1016/0022-4049(80)90096-1, (publisher pdf)