nLab
category of G-sets

Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

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

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

Definition

Let GG be a topological group.

Definition

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

There is a simple characterization of when a GG-action is continuous.

Proposition

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

Proof

See G-sets.

Properties

Notation

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

Proposition

The inclusion i G:GSetG δSeti_G: G Set \to G^\delta Set has a right adjoint r G:G δGSetr_G: G^\delta \to G Set.

Proof

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

Proposition

The forgetful functor U:G δSetSetU: 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 GG-action we can put on the limits or colimits of the underlying sets of the GG-sets.

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

Proposition

The inclusion i G:GSetG δSeti_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 δSetSetU: 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 GSetG Set has all finite limits and arbitrary colimits.

Corollary

The adjunction i Gr Gi_G \dashv r_G is a geometric morphism.

Corollary

A map in GSetG 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 SetSet preserves all finite limits and colimits.

Corollary

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

Proposition

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

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

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

Y X=r G(i G(Y) i G(X)). Y^X = r_G(i_G(Y)^{i_G(X)}).
Proof

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

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

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

(gf^(a))(x)=gf^(a)(g 1x)=gf(a,g 1x)=f(ga,x)=f^(ga)(x). (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

gf^(a)=f^(ga). g \cdot \hat{f}(a) = \hat{f}(g \cdot a).

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

gf(a,x)=gf^(a)(x)=(gf^(a))(gx)=f^(ga)(gx)=f(ga,gx). 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 ff is G δG^\delta-invariant. It is clear that these operations are inverses to each other, and straightforward computations show that this is natural in AA, XX and YY.

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

GSet(A,r G(i G(Y) i G(X))) =G δSet(i G(A),i G(Y) i G(X)) =G δSet(i G(A)×i G(X),i G(Y)) =G δSet(i G(A×X),i G(Y)) =GSet(A×X,Y),\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))=Xr_G(i_G(X)) = X for all XGSetX \in G Set.

Corollary

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

Theorem

The category GSetG Set is an elementary topos.

Equivalent characterizations

As a Grothendieck topos

Theorem

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

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

More generally, by the comparison lemma, we have

Theorem

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

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

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

As a comonad algebra

To be included.

As a classifying topos

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 GG in a topos \mathcal{E}, we write the resulting topos as GG\mathcal{E}.

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

Proposition

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

Proof sketch

This is a straightforward computation. Given an XH(GSet)X \in H (G Set), we write μ:H×XX\mu: H \times X \to X for the action map, and the action of GG merely by a dot. Then we define the action of HGH \rtimes G is given by

((h,g),x)μ(h,gx). ((h, g), x) \mapsto \mu(h, g \cdot x).

Conversely, given an object XX with an HGH \rtimes G action, we give it a GG action by gx=(e,g)xg \cdot x = (e, g) \cdot x, and an HH-action by μ(h,x)=(h,e)x\mu(h, x) = (h, e) \cdot x.

The case of topological groups is more complicated, because an internal topology TT on a space XX is an internally complete lattice TP(X)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 XX an external topological space. Then we have the following result:

Proposition

Let GG be a topological group, and let HH be a topological group object in GSetG Set. Then H(GSet)H (G Set) is equivalent to (HG)Set(H \rtimes G) Set, where HGH \rtimes G is given the product topology.

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

References

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

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

Last revised on October 3, 2018 at 15:49:17. See the history of this page for a list of all contributions to it.