nLab geometry of physics -- representations and associated bundles

This entry contains one chapter of geometry of physics. See there for background and context.

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Representations and associated bundles

The mathematical term group is short for group of symmetries, namely of symmetries of some object. That a group $G$ is the group of symmetries of some $V$ is technically expressed by there being an action of $G$ on $X$. Generally, or at least if $V$ and this representation are suitably linear, this is also called a representation of $G$ (namely a representation of the abstract group as an actual group of symmetries).

The chapter geometry of physics – groups discusses in detail how in geometric homotopy theory groups $G$ are equivalent to groupoids $\mathbf{B}G$ which have a single object and $G$ as the automorphisms of that object, if only $\mathbf{B}G$ is regarded as a pointed object via the canonical base point inclusion $\ast \to \mathbf{B}G$.

This perspective turns out to be exceedingly useful for the discussion of the representations of $G$: these turn out to be equivalent simply to $\mathbf{B}G$-dependent types, hence to any bundles $E \to \mathbf{B}G$ over $G$. Given such, then the object that $V$ that this encodes an action on is the homotopy fiber of this map

$$\array{ V &\longrightarrow& E \\ && \downarrow \\ && \mathbf{B}G } \,.$$

In traditional literature this is familiar in special cases, where the perspective is usually the opposite: given an action of $G$ on $V$, then there is the associated bundle $E = \mathbf{E}G \times_G V$ which is associated to the $G$-universal principal bundle via the action.

Indeed, in the generality of geometric homotopy theory, this association is an equivalence, so that actions and universal associated bundles are essentially the same concept.

Model Layer

1-Representations of 1-Groups

We discuss here ordinary groups (i.e. infinity-groups which are just 1-groups), and their ordinary actions and ordinary associated bundles. Even that ordinary case profits from its formulation via action groupoids, but its key advantage is that this formulation seamlessly generalizes.

Actions

We discuss here traditional concept of discrete groups acting on a sets (“permutation representations”) but phrased in terms of action groupoids sliced over delooping groupoids. The discussion immediately, and essentially verbatim, generalizes to pre-smooth groupoids and to smooth groupoids proper.

Write Grpd for the (2,1)-category of groupoids, the full sub-(infinity,1)-category of ∞Grpd on the 1-truncated objects.

We write

$$X_\bullet = (X_1 \stackrel{\longrightarrow}{\longrightarrow} X_0)$$

for a groupoid object given by an explicit choice of set of objects and of morphisms and then write $X \in Grpd$ for the object that this presents in the $(2,1)$-category. Given any such $X$, we recover a presentation by choosing any essentially surjective functor $S \to X$ (an atlas) out of a set $S$ (regarded as a groupoid) and setting

$$X_\bullet = (S \underset{X}{\times} S \stackrel{\longrightarrow}{\longrightarrow} S)$$

hence taking $S$ as the set of objects and the homotopy fiber product of $S$ with itself over $X$ as the set of morphism.

For $G$ a discrete group, then $\mathbf{B}G$ denotes the groupoid presented by $(\mathbf{B}G)_\bullet = (G \stackrel{\longrightarrow}{\longrightarrow}\ast)$ with composition operation given by the product in the group. Of the two possible ways of making this identification, we agree to use

$$\array{ && \ast \\ & {}^{\mathllap{g_1}}\nearrow && \searrow^{\mathrlap{g_2}} \\ \ast && \underset{g_1 \cdot g_2}{\longrightarrow} && \ast } \,.$$

Definition

Given a discrete group $G$ and an action $\rho$ of $G$ on a set $S$

$$\rho \colon S \times G \longrightarrow S$$

then the corresponding action groupoid is

$$(S//G)_\bullet \coloneqq \left( S\times G \stackrel{\overset{p_1}{\longrightarrow}}{\underset{\rho}{\longrightarrow}} S \right)$$

with composition given by the product in $G$. Hence the objects of $S$ are the elements of $S$, and the morphisms $s \stackrel{}{\longrightarrow } t$ are labeled by elements $g\in G$ and are such that $t = \rho(s)(g)$.

Schematically:

$$(S//G)_\bullet = \left\{ \array{ && \rho(s)(g) \\ & {}^{\mathllap{g_1}}\nearrow && \searrow^{\mathrlap{g_2}} \\ s && \underset{g_1 g_2}{\longrightarrow} && \rho(s)(g_1 g_2) } \right\} \,.$$

Example

For the unique and trivial $G$-action on the singleton set $\ast$, we have

$$\ast//G \simeq \mathbf{B}G \,.$$

This makes it clear that:

Proposition

In the situation of def. , there is a canonical morphism of groupoids

$$(p_\rho)_\bullet \;\colon\; (S//G)_\bullet \longrightarrow (\mathbf{B}G)_\bullet$$

which, in the above presentation, forgets the labels of the objects and is the identity on the labels of the morphisms.

This morphism is an isofibration.

Proposition

For $G$ a discrete group, given two $G$-actions $\rho_1$ and $\rho_2$ on sets $S_1$ and $S_2$, respectively, then there is a natural equivalence between the set of action homomorphisms (“intertwiners”) $\rho_1 \to \rho_2$, regarded as a groupoid with only identity morphisms, and the hom groupoid of the slice $Grpd_{/\mathbf{B}G}$ between their action groupoids regarded in the slice via the maps from prop.

$$G Act(\rho_1,\rho_2) \simeq Grpd_{/\mathbf{B}G}(p_{\rho_1}, p_{\rho_2}) \,.$$

Proof

One quick way to see this is to use, via the discussion at slice (infinity,1)-category, that the hom-groupoid in the slice is given by the homotopy pullback of unsliced hom-groupoids

$$\array{ Grpd_{/\mathbf{B}G}(p_{\rho_1}, p_{\rho_2}) &\longrightarrow& Grpd(S_1//G, S_2//G) \\ \downarrow &(pb)& \downarrow^{\mathrlap{Grpd(S_1//G,p_{\rho_2})}} \\ \ast &\stackrel{}{\longrightarrow}& Grpd(S_1//G, \mathbf{B}G) } \,.$$

Now since $(p_{\rho_2})_\bullet$ is an isofibration, so is $Grpd((S_1//G)_\bullet, (p_{\rho_2})_\bullet)$, and hence this is computed as an ordinary pullback (in the above presentation). That in turn gives the hom-set in the 1-categorical slice. This consists of functors

$$\phi_\bullet \colon (S_1//G)_\bullet \longrightarrow (S_1//G)_\bullet$$

which strictly preserves the $G$-labels on the morphisms. These are manifestly the intertwiners.

$$\phi_\bullet \;\colon\; \left( \array{ s \\ \downarrow^{\mathrlap{g}} \\ \rho(s)(g) } \right) \mapsto \left( \array{ \phi(s) \\ \downarrow^{\mathrlap{g}} \\ \phi(\rho(s)(g)) & = \rho(\phi(s))(g) } \right) \,.$$

Proposition

The homotopy fiber of the morphism in prop. is equivalent to the set $S$, regarded as a groupoid with only identity morphisms, hence we have a homotopy fiber sequence of the form

$$\array{ S &\longrightarrow& S//G \\ && \downarrow^{\mathrlap{p_\rho}} \\ && \mathbf{B}G } \,.$$

Proof

In the presentation $(S//G)_\bullet$ of def. , $p_\rho$ is an isofibration, prop. . Hence the homotopy fibers of $p_\rho$ are equivalent to the ordinary fibers of $(p_\rho)_\bullet$ computed in the 1-category of 1-groupoids. Since $(p_\rho)_\bullet$ is the identity on the labels of the morphisms in this presentation, this ordinary fiber is precisely the sub-groupoid of $(S//G)_\bullet$ consisting of only the identity morphismss, hence is the set $S$ regarded as a groupoid.

Conversely, the following construction extract a group action from a homotopy fiber sequence of groupoids of this form.

Definition

Given a homotopy fiber sequence of groupoids of the form

$$\array{ S &\stackrel{i}{\longrightarrow}& E \\ && \downarrow^{\mathrlap{p}} \\ && \mathbf{B}G }$$

such that $S$ is equivalent to a set $S$, define a $G$-action on this set as follows.

Consider the homotopy fiber product

$$S \underset{E}{\times} S \stackrel{\overset{}{\longrightarrow}}{\underset{}{\longrightarrow}} S$$

of $i$ with itself. By the pasting law applied to the total homotopy pullback diagram

$$\array{ S \underset{E}{\times} S &\longrightarrow& S \\ \downarrow && \downarrow^{\mathrlap{i}} \\ S &\stackrel{i}{\longrightarrow}& E \\ \downarrow && \downarrow^{\mathrlap{p}} \\ \ast &\longrightarrow& \mathbf{B}G } \;\;\;\; \simeq \;\;\;\; \array{ S\times G &\stackrel{p_1}{\longrightarrow}& S \\ \downarrow && \downarrow \\ G &\stackrel{}{\longrightarrow}& \ast \\ \downarrow && \downarrow \\ \ast &\longrightarrow& \mathbf{B}G }$$

there is a canonical equivalence of groupoids

$$S \underset{E}{\times} S \simeq S \times G$$

such that one of the two canonical maps from the fiber product to $S$ is projection on the first factor. The other map under this equivalence we denote by $\rho$:

$$S \times G \stackrel{\overset{p_1}{\longrightarrow}}{\underset{\rho}{\longrightarrow}} S \,.$$

Remark

The functor $i \colon S \to E$ is clerly essentially surjective (every connected component of $E$ has a homotopy fiber under its map to $\mathbf{B}G$). This implies that $E$ is presented by

$$E_\bullet \coloneqq (S \underset{E}{\times}S \stackrel{\overset{p_1}{\longrightarrow}}{\underset{p_2}{\longrightarrow}} S)$$

and hence, via the construction in def. , by

$$E_\bullet \simeq (S \times G \stackrel{\overset{p_1}{\longrightarrow}}{\underset{\rho}{\longrightarrow}} S) \,.$$

But this already exhibits $E$ as an action groupoid, in particular it mans that $\rho$ is really an action:

Proposition

The morphism $\rho$ constructed in def. is a $G$-action in that it satisfies the action propery, which says that the diagram (of sets)

$$\array{ S\times G \times G &\stackrel{(id,(-)\cdot(-))}{\longrightarrow}& S \times G \\ \downarrow^{\mathrlap{(\rho,id)}} && \downarrow^{\mathrlap{\rho}} \\ S \times G &\stackrel{\rho}{\longrightarrow}& S }$$

commutes.

Proposition

For $G$ a discrete group, there is an equivalence of categories

$$G Act(Set) \stackrel{\simeq}{\longrightarrow} (Grpd_{/\mathbf{BG}})_{\leq 0}$$

between the category of permutation representations of $G$ and the full subcategory of the slice (2,1)-category of Grpd over $\mathbf{B}G$ on the 0-truncated objects.

This equivalence takes an action to its action groupoid.

Proof

By remark the construction of action groupoids is essentially surjective. By prop. it is fully faithful.

Examples of actions

One remarkable consequence of prop. is that it says that categories of actions are slices of (2,1)-toposes, hence are slice (2,1)-toposes hence in particular are themselves (2,1)-topos. In particular there is an internal hom of actions. This is the conjugation action construction.

Definition

Given a discrete group $G$ and two $G$-actions $\rho_1$ and $\rho_2$ on sets $S_1$ and $S_2$, respectively, then the function set $[S_1, S_2]$ is naturally equipped with the conjugation action

$$Ad \;\colon \; [S_1, S_2] \times G \longrightarrow [S_1,S_2]$$

which takes $((S_1 \stackrel{f}{\to} S_2), g)$ to

$$\rho_2(-)(g)\circ f \circ \rho_1(-)(g^{-1}) \;\colon\; S_1 \stackrel{\rho_1(-)(g^{-1})}{\longrightarrow} S_1 \stackrel{f}{\longrightarrow} S_2\stackrel{\rho_2(-)(g)}{\longrightarrow} S_2 \,.$$

Proposition

The conjugation action construction of def. is the internal hom in the category of actions.

Proof

We need to show that for any three permutation representations, functions

$$\phi \;\colon\; S_3 \longrightarrow [S_1,S_2]$$

which intertwine the $G$-action on $S_3$ with the conjugation action on $[S_1,S_2]$ are in natural bijection with functions

$$\tilde \phi \;\colon\; S_3 \times S_1 \longrightarrow S_2$$

which intertwine the diagonal action on the Cartesian product $S_3 \times S_1$ with the action on $S_2$.

The condition on $\phi$ means that for all $g\in G$ and $s_3 \in S_3$ it sends

$$\phi \;\colon\; \rho_3(s_3)(g) \mapsto \left( s_1 \mapsto \rho_2\left( \phi\left(s_3\right)\left( \rho_1\left(s_1\right)\left(g^{-1}\right) \right)\right)\left(g\right) \right) \,.$$

This is equivalently a function $\tilde \phi$ of two variables which sends

$$\tilde \phi \;\colon\; (\rho_3(s_3)(g), s_1) \mapsto \rho_2 ( \phi(s_3)( \rho_1(s_1)(g^{-1}) ) )(g) \,.$$

Since this has to hold for all values of the variables, it has to hold when substituing $s_1$ with $\rho_1(s_1)(g)$. After this substitution the above becomes

$$\tilde \phi \;\colon\; (\rho_3(s_3)(g), \rho_1(s_1)(g)) \mapsto \rho_2(\phi(s_3)(s_1 ))(g) \,.$$

This is the intertwining condition on $\tilde \phi$. Conversely, given $\tilde \phi$ satisfying this for all values of the variables, then running the argument backwards shows that its hom-adjunct $\phi$ satisfies its required intertwining condition.

The following is immediate but conceptually important:

Proposition

The invariants of the conjugation action on $[S_1,S_2]$ is the set of action homomorphisms/intertwiners.

Hence the inclusion of invariants into the conjugation action gives the inclusion of the external hom set of the category of $G$-actions into the set underlying the internal hom

$$G Act(\rho_1,\rho_2)\hookrightarrow [\rho_1,\rho_2] \,.$$

Example

Given any $X$ with its canonical action of its automorphism group $Aut(X)$, regard any $Y$ as equipped with the trivial $Aut(Y)$-action.

Then the conjugation action, def. , on $[X,Y]$ is the action by precomposition with automorphisms of $X$.

Associated bundles

At geometry of physics – principal bundles in the section Smooth principal bundles via smooth groupoids is discussed how smooth principal bundles for a Lie group $G$ over a smooth manifold $X$ are equivalently the homotopy fibers of morphisms of smooth groupoids (smooth stacks) of the form

$$X \stackrel{}{\longrightarrow} \mathbf{B}G \,.$$

Now given an action $\rho$ of $G$ on some smooth manifold $V$, and regardiing this action via its action groupoid projection $p_\rho \colon V//G \to \mathbf{B}G$ as discussed above, then we may consider these two morphisms into $\mathbf{B}G$ jointly

$$\array{ && V//G \\ && \downarrow^{\mathrlap{p_\rho}} \\ X &\stackrel{g}{\longrightarrow}& \mathbf{B}G }$$

and so it is natural to construct their homotopy fiber product.

We now discuss that this is equivalently the associated bundle which is associated to the principal bundle $P \to X$ via the action $\rho$.

Proposition

For $G$ a smooth group (e.g. a Lie group), $X$ a smooth manifold, $P \to X$ a smooth $G$-principal bundle over $X$ and $\rho$ a smooth action of $G$ on some smooth manifold $V$, then the associated $V$-fiber bundle $P \times_G V\to X$ is equivalently (regarded as a smooth groupoid) the homotopy pullback of the action groupoid-projection $p_\rho \colon V//G \to \mathbf{B}G$ along a morphism $g \colon X\to\mathgbf{B}G$ which modulates $P$

$$\array{ P\times_G V &\longrightarrow& V//G \\ \downarrow && \downarrow^{\mathrlap{p_\rho}} \\ X &\stackrel{g}{\longrightarrow}& \mathbf{B}G } \,.$$

Proof

By the discussion at geometry of physics – principal bundles in the section Smooth principal bundles via smooth groupoids, the morphism $g$ of smooth groupoids is presented by a morphism of pre-smooth groupoids after choosing an open cover $\{U_i \to X\}$ over wich $P$ trivialize and choosing a trivialization, by the zig-zag

$$\array{ C(\{U_i\})_\bullet &\stackrel{g_\bullet}{\longrightarrow}& (\mathbf{B}G)_\bullet \\ \downarrow^{\mathrlap{\simeq_{lwe}}} \\ X }$$

where the top morphism is the Cech cocycle of the given local trivialization regarded as a morphism out of the Cech groupoid of the given cover.

Moreover, by prop. the morphism $(p_\rho)_\bullet$ is a global fibration of pre-smooth groupoids, hence, by the discussion at geometry of physics – smooth homotopy types, the homotopy pullback in question is equivalently computed as the ordinary pullback of pre-smooth groupoids of $(p_\rho)_\bullet$ along this $g_\bullet$

$$\array{ C(\{U_i\})_\bullet \underset{(\mathbf{B}G)_\bullet}{\times} (V//G)_\bullet &\longrightarrow& (V//G)_\bullet \\ \downarrow && \downarrow^{\mathrlap{(p_\rho)_\bullet}} \\ C(\{U_i\})_\bullet &\stackrel{g_\bullet}{\longrightarrow}& (\mathbf{B}G)_\bullet \\ \downarrow^{\mathrlap{\simeq_{lwe}}} \\ X } \,.$$

This pullback is computed componentwise. Hence it is the pre-smooth groupoid whose morphisms are pairs consisting of a morphism $(x,i)\to (x,j)$ in the Cech groupoid as well as a morphism $s \stackrel{g}{\to} \rho(s)(g)$ in the action groupoid, such that the group label $g$ of the latter equals the cocycle $g_{i j}(x)$ of the cocycle on the former. Schematically:

$$C(\{U_i\})_\bullet \underset{(\mathbf{B}G)_\bullet}{\times} (V//G)_\bullet = \left\{ ((x,i),s) \stackrel{g_{i j}(x)}{\longrightarrow} ((x,j),\rho(s)(g)) \right\} \,.$$

This means that the pullback groupoid has at most one morphism between every ordered pair of objects. Accordingly this groupoid is equivalence of groupoids equivalent to the quotient of its space of objects by the equivalence relation induced by its morphisms:

$$\cdots \simeq \left( \underset{i}{\coprod} U_i \times V \right)/_\sim \,.$$

This is a traditional description of the associated bundle in question.

Invariants and sections

One advantage of the perspective on representations via action groupoids is that it gives a good formulation of the invariants and the coinvariants of actions. The invariants are the sections of the action groupoid projection, while the coivariants in fact are the action groupoid itself.

Proposition

For $G$ a discrete group, $\rho$ a $G$-action on some set $S$, then the set of invariants of that action is equivalent to the groupoid of sections of the action groupoid projection of prop. , corresponding to the action via prop. .

Proof

The sections in question are diagrams in Grpd of the form

$$\array{ \mathbf{B}G && \stackrel{\sigma}{\longrightarrow} && S//G \\ & {}_{\mathllap{id}}\searrow &\swArrow_{\mathrlap{\simeq}}& \swarrow_{\mathrlap{p_\phi}} \\ && \mathbf{B}G } \,,$$

hence the groupoid which they form is equivalently the hom-groupoid

$$Grpd_{/\mathbf{B}G}(id_{\mathbf{B}G}, p_\rho) \in Grpd$$

in the slice of Grpd over $\mathbf{B}G$. As in the proof of prop. , with the fibrant presentation $(p_\rho)_\bullet$ of prop. , this is equivalently given by strictly commuting diagrams of the form

$$\array{ (\mathbf{B}G)_\bullet && \stackrel{\sigma_\bullet}{\longrightarrow} && (S//G)_\bullet \\ & {}_{\mathllap{id_\bullet}}\searrow &=& \swarrow_{\mathrlap{(p_\phi)_\bullet}} \\ && (\mathbf{B}G)_\bullet } \,.$$

These $\sigma$ now are manifestly functors that are the identiy on the group labels of the morphisms

$$\sigma_\bullet \;\colon\; \left( \array{ \ast \\ \downarrow^{\mathrlap{g}} \\ \ast } \right) \;\; \mapsto \;\; \left( \array{ \sigma(\ast) \\ \downarrow^{\mathrlap{g}} \\ \sigma(\ast) & = \rho(\sigma(\ast)(g)) } \right) \,.$$

This shows that they pick precisely those elements $\sigma(\ast) \in S$ which are fixed by the $G$-action $\rho$.

Moreover, since these functors are identity on the group labels, there are no non-trivial natural isomorphisms between them, and hence the groupoid of sections is indeed a set, the set of invariant elements.

More generally, we may consider sections of these groupoid projections after pulling them back along some cocycle:

Proposition

Given an associated bundle $P \times_G V\to X$ modulated, as in prop. , by a morphism of smooth groupoids of the form $g \colon X \longrightarrow \mathbf{B}G$, then its set of sections is equivalently the groupoid of diagrams

$$\array{ X && \stackrel{\sigma}{\longrightarrow} && S//G \\ & {}_{\mathllap{g}}\searrow &\swArrow_{\mathrlap{\simeq}}& \swarrow_{\mathrlap{p_\phi}} \\ && \mathbf{B}G } \,,$$

hence the groupoid of sections is equivalently the slice hom-groupoid

$$\Gamma_X(P\times_G V) \simeq Grpd_{/\mathbf{B}G}(g, p_\rho) \,.$$

Proof

By the defining universal property of the homotopy pullback in prop. .

Remark

Taken together this means that invariants of group actions are equivalently the sections of the corresponding universal associated bundle.

Stabilizer groups

Given an ordinary discrete group $G$ and an action $\rho \colob G \times V \longrightarrow V$ of the group on some set $V$, then for $x \in V$ any element, the stabilizer group

$$Stab_x(V) \hookrightarrow G$$

is the subgroup of $G$ consisting of those group elements under whose action the element $v$ does not change (is “stable”):

$$Stab_G(x) = \left\{ g \in G \;|\; \rho(g)(v) = v \right\} \,.$$

We observe that this has the following intrinsic reformulation. Write again

$$\array{ V &\stackrel{i}{\longrightarrow}& V//G \\ && \downarrow^{\mathrlap{p_\rho}} \\ && \mathbf{B}G }$$

for the action groupoid homotopy fiber sequence that corresponds to $\rho$ via prop. .

Proposition

The stabilizer group of $x$ under $G$ acting on $V$ is equivalently the looping of the action groupoid at the point $i(x)$:

$$Stab_G(x) \simeq \Omega_{i(x)} (V//G)$$

Proof

Since $V//G$ is a 1-groupoid, the loop space object $\Omega_{i(x)} (V//G)$ is simply the automorphism group of $x$ regarded as an object in the action groupoid, hence the group of morphisms in $(V//G)_\bullet$ from $x$ to $x$. Comparison with def. shows that these morphisms are precisely those labeled by elements $g \in G$ for which $\rho(g)(x) = x$.

$\infty$-Representations of 1-groups

The above perspective on ordinary representations of ordinary groups on sets via their action groupoid projection has the advantage that it immediately generalizes to a definition where 1-groups act on more general homotopy types up to coherent homotopy, hence to infinity-representations or infinity-actions.

Definition

Given a discrete group $G$ and a Kan complex $V_\bullet$, then an infinity-representation or infinity-action of $G$ on $V$ is another Kan complex, to be denoted $(V//G)_\bullet$, equipped with a simplicial map $(p_\rho) \colon (V//G)_\bullet \longrightarrow N(\mathbf{B}G)_\bullet$ to the nerve of $(\mathbf{B}G_\bullet)$, such that the homotopy fiber of that map is weakly homotopy equivalent to $V_\bullet$.

Examples of $\infty$-Representations

Given an abelian group $A$ and $n \in \mathbb{N}$, write $(\mathbf{B}^n A)_\bullet$ for the Kan complex which is the image under the Dold-Kan correspondence of the chain complex that is concentrated on $A$ in degree $n$.

Then for $G$ a discrete group, the mapping complex

$$[\mathbf{B}G,\mathbf{B}^n A] \in KanCplx$$

is the infinity-groupoid whose objects are the degree-$n$ group cocycles on $G$ with coefficients in $A$ (regarded as a $G$-module with trivial action), whose morphisms are the coboundaries between these cocycles, and whose higher morphisms are higher order coboundaries-of-coboundaries.

Being a mapping space, this naturally carries a precomposition action by the automorphism infinity-group of $\mathbf{B}G$, which is also known as the automorphism 2-group of $G$. Restricting this to pointed automorphisms is is the 1-group $Aut_{Grp}(G)$ of invertible group homomorphisms of $G$.

Semantic Layer

We now consider any (∞,1)-topos $\mathbf{H}$ and formulate the group actions and their associated bundles in general abstractly here.

By the discussion at geometry of physics – groups, group objects in an (∞,1)-category are equivalently pointed connected objects which we write $(\ast\to\mathbf{B}G)\in \mathbf{H}^{\ast/}$, the group itself being the loop space object

$$G \simeq \Omega \mathbf{B}G$$

formed at the given base point.

$\infty$-Actions

In view of this, the characterization of ordinary group action according to prop. has an immediate generalization to ∞-groups in any (∞,1)-topos

Definition

Given an ∞-group $G \in \mathbf{H}$ and any object $V \in \mathbf{H}$, then an ∞-action $\rho$ of $G$ on $V$ is a homotopy fiber sequence in $\mathbf{H}$ of the form

$$\array{ V &\to& V\sslash G \\ && \downarrow^{\mathrlap{p_\rho}} \\ && \mathbf{B}G } \,.$$

The object $V/G$ defined thereby we call the homotopy quotient of $V$ by $G$ via this action.

Associated $\infty$-bundles

• associated infinity-bundle

$$\array{ E &\to& V\sslash G \\ \downarrow &pb& \downarrow \\ \tilde X &\to& \mathbf{B}G \\ \downarrow^{\mathrlap{\simeq}} \\ X }$$

• section

$$\array{ X &&\stackrel{\sigma}{\to}&& V \sslash G \\ & \searrow &\swArrow_{\simeq}& \swarrow \\ && \mathbf{B}G }$$

Stabilizer $\infty$-Groups

We had seen above in prop. that the traditional concept of stabilizer groups of group actions is equivalent to groups of loops in the action groupoid of the given action. This equivalent reformulation has an immediate generalization to ∞-actions, def. .

For $\mathbf{H}$ an (∞,1)-topos, $G\in \mathbf{H}$ an object equipped with ∞-group structure, hence with a delooping $\mathbf{B}$G, and for $\rho$ an ∞-action of $G$ on some $V$, exhibited by a homotopy fiber sequence of the form

$$\array{ V &\stackrel{i}{\longrightarrow}& V/G \\ && \downarrow^{\mathrlap{p_\rho}} \\ && \mathbf{B}G } \,.$$

Definition

Given a global element of $V$

$$x \colon \ast \to X$$

then the stabilizer $\infty$-group $Stab_\rho(x)$ of the $G$-action at $x$ is the loop space object

$$Stab_\rho(x) \coloneqq \Omega_{i(x)} (X/G) \,.$$

Remark

Equivalently, def. , gives the loop space object of the 1-image $\mathbf{B}Stab_\rho(x)$ of the morphism

$$\ast \stackrel{x}{\to} X \to X/G \,.$$

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. to a diagram of the form

$$\array{ \ast &\stackrel{x}{\longrightarrow}& X &\stackrel{}{\longrightarrow}& X/G \\ \downarrow^{\mathrlap{epi}} && & \nearrow_{\mathrlap{mono}} & \downarrow \\ \mathbf{B} Stab_\rho(x) &=& \mathbf{B} Stab_\rho(x) &\longrightarrow& \mathbf{B}G } \,.$$

In particular there is hence a canonical homomorphism of $\infty$-groups

$$Stab_\rho(x) \longrightarrow G \,.$$

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.

Syntactic Layer

The context of a pointed connected type: representation theory

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Dependent product over a pointed connected type: invariants

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Dependent sum over a pointed connected type: quotients

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