Cohomology and Extensions
Given an action of a group on a set , for every element , the stabilizer subgroup of x (also called the isotropy group of ) is the set of all elements in that leave fixed:
If all stabilizer groups are trivial, then the action is called a free action.
General abstract characterization
We discuss stabilizer subgroups from the nPOV.
A group action is equivalently encoded in its action groupoid fiber sequence in Grpd
where the is the action groupoid itself, is the delooping groupoid of and is regarded as a 0-truncated groupoid.
This fiber sequence may be thought of as being the -associated bundle to the -universal principal bundle. (Here discussed for a discrete group but this discussion goes through verbatim for a cohesive group).
any global element of , we have an induced element of the action groupoid and may hence form the first homotopy group . This is the stabilizer group. Equivalently this is the loop space object of at , given by the homotopy pullback
This characterization immediately generalizes to stabilizer ∞-groups of ∞-group actions. This we discuss below
For -group actions
Let be an (∞,1)-topos and be an ∞-group object in . Write for its delooping object.
Then for any other object, an action of on is an object and a fiber sequence of the form
The action as a morphism is recovered from this by the (∞,1)-pullback
Now for any global element, the stabilizer -group of at is the loop space object
This is equipped with a canonical morphism of ∞-group objects
given by the looping of
For an -group acting on itself
For any ∞-group in an (∞,1)-topos , its (right) action on itself is given by the looping/delooping fiber sequence
Clearly, for every point we have is trivial. Hence the action is free.
Stabilizers of shapes
For an action, and any other object, we get an induced action on the internal hom defined as the (∞,1)-pullback
where the bottom morphism is the internal hom adjunct of the projection .
Then for a “shape” in , the stabilizer ∞-group of under is .
The morphism of -groups
characterizes the higher Klein geometry induced by .