geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Given an action of a group on some space, and given a point or (or more generally some subspace), then the stabilizer group of that point (that subspace) is the subgroup whose action leaves the point (the subspace) fixed, invariant.
The importance of stabilizer subgroups for the general development of geometry was famously highlighted in (Klein 1872) in the context of what has come to be known the Erlangen program. For more on this aspect see at Klein geometry and Cartan geometry.
Given an action $G\times X\to X$ of a group $G$ on a set $X$, for every element $x \in X$, the stabilizer subgroup of $x$ (also called the isotropy group of $x$) is the set of all elements in $G$ that leave $x$ fixed:
If all stabilizer groups are trivial, then the action is called a free action.
We reformulate the traditional definition above from the nPOV, in terms of homotopy theory.
A group action $\rho\colon G \times X \to X$ is equivalently encoded in its action groupoid fiber sequence in Grpd
where the $X/G$ is the action groupoid itself, $\mathbf{B}G$ is the delooping groupoid of $G$ and $X$ is regarded as a 0-truncated groupoid.
This fiber sequence may be thought of as being the $\rho$-associated bundle to the $G$-universal principal bundle. (Here discussed for $G$ a discrete group but this discussion goes through verbatim for $G$ a cohesive group.)
For
any global element of $X$, we have an induced element $x\colon * \to X \to X/G$ of the action groupoid and may hence form the first homotopy group $\pi_1(X/G, x)$. This is the stabilizer group. Equivalently this is the loop space object of $X/G$ at $x$, given by the homotopy pullback
This characterization immediately generalizes to stabilizer ∞-groups of ∞-group actions. This we discuss below
Let $\mathbf{H}$ be an (∞,1)-topos and $G \in \infty Grp(G)$ be an ∞-group object in $\mathbf{H}$. Write $\mathbf{B}G \in \mathbf{H}$ for its delooping object.
By the discussion at ∞-action we have the following.
For $X \in \mathbf{H}$ any object, an ∞-action of $G$ on $X$ is equivalently an object $X/G$ and a homotopy fiber sequence of the form
Here $X/G$ is the homotopy quotient of the ∞-action
The action as a morphism $X \times G \to X$ is recovered from prop. 1 by the (∞,1)-pullback
Given an ∞-action $\rho$ of $G$ on $X$ as in prop. 1, and given a global element of $X$
then the stabilizer $\infty$-group $Stab_\rho(x)$ of the $G$-action at $x$ is the loop space object
Equivalently, def. 1, gives the loop space object of the 1-image $\mathbf{B}Stab_\rho(x)$ of the morphism
As such the delooping of the stabilizer $\infty$-group sits in a 1-epimorphism/1-monomorphism factorization $\ast \to \mathbf{B}Stab_\rho(x) \hookrightarrow X/G$ which combines with the homotopy fiber sequence of prop. 1 to a diagram of the form
In particular there is hence a canonical homomorphism of $\infty$-groups
However, in contrast to the classical situation, this morphism is not in general a monomorphism anymore, hence the stabilizer $Stab_\rho(x)$ is not a sub-group of $G$ in general.
For $G$ any ∞-group in an (∞,1)-topos $\mathbf{H}$, its (right) action on itself is given by the looping/delooping fiber sequence
Clearly, for every point $g \in G$ we have $Stab_{\rho}(g) \simeq * \times_* * \simeq *$ is trivial. Hence the action is free.
Let $X \to X/G \stackrel{\rho}{\to} \mathbf{B}G$ be an ∞-action of $G$ on $X$.
Let $Y \in \mathbf{H}$ any other object, and regard it as equipped with the trivial $G$-action $Y \to Y \times \mathbf{B}G \to \mathbf{B}G$. There is then an induced ∞-action $\rho_Y$ on the internal hom $[Y,X]$, the conjugation action, given by internal hom in the slice (∞,1)-topos over $\mathbf{B}G$:
Now given any $f \colon Y \to X$, then the stabilizer group $Stab_{\rho_Y}(f)$ is the stabilizer of $Y$ “in” $X$ under this $G$-action.
The morphism of $\infty$-groups
hence characterizes the (higher) Klein geometry induced by the $G$-action and by the shape $f\colon Y \to X$. (See at Klein geometry – History.)
For completeness we notice that:
Here $(\underset{\mathbf{B}G}{\sum} [Y,X]_{/\mathbf{B}G} \to \mathbf{B}G )$ is equivalently the (∞,1)-pullback $\rho_Y$ in
where the bottom morphism is the internal hom adjunct of the projection $Y \times \mathbf{B}G \to \mathbf{B}G$.
We check the Hom adjunction property, that for any $(A/G \stackrel{\alpha}{\to} \mathbf{B}G) \in \mathbf{B}G$ we have
with $[Y,X]_{/\mathbf{B}G}$ replaced by the above pullback.
Notice that by the $G$-action on $Y$ being trivial, we have $A \times_{\mathbf{B}G} Y \simeq (A/G \times Y \stackrel{p_1}{\to} A/G \stackrel{\alpha}{\to} \mathbf{B}G) \in \mathbf{H}_{/\mathbf{B}G}$.
Then use the characterization of Hom-spaces in a slice to find $\mathbf{H}_{/\mathbf{B}G}(A,[Y,X]_{\mathbf{B}G})$ as the homotopy pullback on the left of
Now using the Hom-adjunction in $\mathbf{H}$ itself, the fact that $\mathbf{H}(A/G,-)$ preserves homotopy pullbacks and the pasting law this is equivalent to
Here the bottom map is indeed the name of $\alpha \circ p_1$ and so again by the pullback characterization of Hom-spaces in a slice this pasting diagram does exhibit $\mathbf{H}_{/\mathbf{B}G}(A\times_{\mathbf{B}G} Y, X)$ as indicated.
Conversely, for any homomorphism $H \to G$ of ∞-groups given, then the canonical $G$-$\infty$-action for which $H$ is the stabilizer $\infty$-group of a point is the canonical action on (the “coset”) $G/H$.
This follows from def. 1 by observing that the homotopy fiber sequence of prop. 1 for the $G$-action on $G/H$ is just
so that for any point $x \colon \ast \to G/H$ we have
Dually to “stabilizers of shapes”, as above one may consider stabilizers of “co-shapes”.
I.e. given a $G$-action on $X$, and given a map $f \colon X \to A$, then one may ask for the stabilizer of $f$ in the canonical $G$-action on $[X,A]$.
For instance if $A$ here is $\mathbf{B}^{n}U(1)_{conn}$, and $f \colon X \to \mathbf{B}^n U(1)_{conn}$ is regarded as a prequantum n-bundle ,and $[X,A]$ is replaced by its differential concretification, then these stabilizers are the quantomorphism n-groups.
to be expanded on
An early occurence of the concept of stabilizer subgroups is in p. 4 of
Felix Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen (1872)
translation by M. W. Haskell, A comparative review of recent researches in geometry , trans. M. W. Haskell, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (retyped pdf, retyped pdf, scan of original)
(the “Erlangen program”).