This entry is about the category of continuous G-sets, for a topological group . Continuous -sets are sets with an action of that is continuous when is given the discrete topology.
The category of continuous -sets is a Grothendieck topos, and is closely related to Fraenkel-Mostowski models.
Let be a topological group.
The category of continuous -sets is the category of sets equipped with a continuous -action , where is given the discrete topology, and the morphisms are the -invariant maps. We write the category as . In this page, we always let act on the left.
There is a simple characterization of when a -action is continuous.
Let be a topological group, and be a set with a action . Then the action is continuous if and only if the stabilizer of each element is open.
See G-sets.
For a topological group , we write for the discrete version of . Every -set can be viewed as a set in the obvious way.
The inclusion has a right adjoint such that for , the continuous -set is the subset of consisting of those points with open stabilizer.
This follows from the observation that if is a -invariant function between -sets, then for each , the stabilizer of includes the stabilizer of .
This follows from the general fact that limits and colimits in presheaf categories are computed pointwise.
Alternatively, there is an obvious -action we can put on the limits or colimits of the underlying sets of the -sets.
The creating limits part also comes from the fact that the forgetful functor is monadic.
The inclusion creates all finite limits and all colimits.
This follows from the observation that the finite limits and colimits created by 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.
The category has all finite limits and arbitrary colimits.
The adjunction is a geometric morphism.
A map in is a monomorphism if and only if it is injective; epimorphism if and only if it is surjective.
A map is monic if and only if its kernel pair is an isomoprhism, and similarly for epic, and the forgetful functor to preserves all finite limits and colimits.
The subobject classifier of is , the two-point set with the trivial -action.
The exponential object in is defined by , with the -action given by
The exponential object in is given by the subset of the functions that have an open stabilizer, ie.
Let be -sets. Given a map , we obtain by
We now check that is -invariant — we have
So we get
Conversely, given a , we obtain by , and we have
So is -invariant. It is clear that these operations are inverses to each other, and straightforward computations show that this is natural in , and .
Then in general, if are in fact -sets, then we can compute
using the fact that for all .
The power object of is given by the subsets of that have an open stabilizer.
The category is an elementary topos.
Let be a topological group. Then the category is equivalent to the topos of sheaves on the atomic site , where the objects of are the open subgroups of , and the morphisms are the left cosets such that , and all non-empty sieves are covering.
Alternatively, it is the full subcategory of containing objects of the form , where is an open subgroup.
More generally, by the comparison lemma, we have
Let be a topological group, and be a cofinal set of open subgroups, ie. every open subgroup contains a member of . Then the category is equivalent to the topos of sheaves on the atomic site , where the objects of are the open subgroups in , and the morphisms are the left cosets such that , and all non-empty sieves are covering.
Alternatively, it is the full subcategory of containing objects of the form , where .
In particular, if is a discrete group, then the trivial subgroup itself is a cofinal set of open subgroups. So is the category of sheaves on the category with only one object, whose morphisms are the elements of . This is the usual characterization of as the functor category .
To be included.
To be included.
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 in a topos , we write the resulting topos as .
A particular interesting case is when we consider an internal group in the topos . For a discrete group, an internal group in is a group with a homomorphism . This allows us to form the semidirect product .
Let be a discrete group, and let be a group object in . Then the category is equivalent to .
This is a straightforward computation. Given an , we write for the action map, and the action of merely by a dot. Then we define the action of is given by
Conversely, given an object with an action, we give it a action by , and an -action by .
The case of topological groups is more complicated, because an internal topology on a space is an internally complete lattice , 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 an external topological space. Then we have the following result:
Let be a topological group, and let be a topological group object in . Then is equivalent to , where is given the product topology.
Proof is a straightforward check that the continuity conditions match up.
Some elementary properties of continuous -sets can be found in books such as
The formal correspondence between permutation models of ZFA and toposes of continuous -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.