Contents

topos theory

Contents

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 is the category of sets $X$ equipped with a continuous $G$-action $\mu: G \times X \to X$, where $X$ is given the discrete topology, and the morphisms are the $G$-invariant maps. We write the category as $G Set$. In this page, we always let $G$ act on the left.

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

Proposition

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

See G-sets.

Properties

Notation

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

Proposition

The inclusion $i_G: G Set \to G^\delta Set$ 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.

Proof

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

Proposition

The forgetful functor $U: G^\delta Set \to Set$ 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 isomoprhism, 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.

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

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 $Set^G$.

To be included.

To be included.

Internal group actions

The same construction works for an internal group in an arbitrary topos, and the resulting category is also a topos, by the same proof. In this case, for a group $G$ in a topos $\mathcal{E}$, we write the resulting topos as $G\mathcal{E}$.

A particular interesting case is when we consider an internal group in the topos $G Set$. For $G$ a discrete group, an internal group in $G Set$ is a group $H$ with a homomorphism $G \to \Aut(H)$. This allows us to form the semidirect product $H \rtimes G$.

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 $\mu: H \times X \to X$ for the action map, and the action of $G$ merely by a dot. Then we define the action of $H \rtimes G$ is given 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.

Proof is a straightforward check that the continuity conditions match up.

References

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

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

Last revised on February 14, 2020 at 12:17:42. See the history of this page for a list of all contributions to it.