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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 GG is the group of symmetries of some VV is technically expressed by there being an action of GG on XX. Generally, or at least if VV and this representation are suitably linear, this is also called a representation of GG (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 GG are equivalent to groupoids BG\mathbf{B}G which have a single object and GG as the automorphisms of that object, if only BG\mathbf{B}G is regarded as a pointed object via the canonical base point inclusion *BG\ast \to \mathbf{B}G.

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

V E BG. \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 GG on VV, then there is the associated bundle E=EG× GVE = \mathbf{E}G \times_G V which is associated to the GG-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 =(X 1X 0) 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 XGrpdX \in Grpd for the object that this presents in the (2,1)(2,1)-category. Given any such XX, we recover a presentation by choosing any essentially surjective functor SXS \to X (an atlas) out of a set SS (regarded as a groupoid) and setting

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

hence taking SS as the set of objects and the homotopy fiber product of SS with itself over XX as the set of morphism.

For GG a discrete group, then BG\mathbf{B}G denotes the groupoid presented by (BG) =(G*)(\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

* g 1 g 2 * g 1g 2 *. \array{ && \ast \\ & {}^{\mathllap{g_1}}\nearrow && \searrow^{\mathrlap{g_2}} \\ \ast && \underset{g_1 \cdot g_2}{\longrightarrow} && \ast } \,.
Definition

Given a discrete group GG and an action ρ\rho of GG on a set SS

ρ:S×GS \rho \colon S \times G \longrightarrow S

then the corresponding action groupoid is

(S//G) (S×Gρp 1S) (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 GG. Hence the objects of SS are the elements of SS, and the morphisms sts \stackrel{}{\longrightarrow } t are labeled by elements gGg\in G and are such that t=ρ(s)(g)t = \rho(s)(g).

Schematically:

(S//G) ={ ρ(s)(g) g 1 g 2 s g 1g 2 ρ(s)(g 1g 2)}. (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 GG-action on the singleton set *\ast, we have

*//GBG. \ast//G \simeq \mathbf{B}G \,.

This makes it clear that:

Proposition

In the situation of def. 1, there is a canonical morphism of groupoids

(p ρ) :(S//G) (BG) (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 GG a discrete group, given two GG-actions ρ 1\rho_1 and ρ 2\rho_2 on sets S 1S_1 and S 2S_2, respectively, then there is a natural equivalence between the set of action homomorphisms (“intertwiners”) ρ 1ρ 2\rho_1 \to \rho_2, regarded as a groupoid with only identity morphisms, and the hom groupoid of the slice Grpd /BGGrpd_{/\mathbf{B}G} between their action groupoids regarded in the slice via the maps from prop. 1

GAct(ρ 1,ρ 2)Grpd /BG(p ρ 1,p ρ 2). 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

Grpd /BG(p ρ 1,p ρ 2) Grpd(S 1//G,S 2//G) (pb) Grpd(S 1//G,p ρ 2) * Grpd(S 1//G,BG). \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 ρ 2) (p_{\rho_2})_\bullet is an isofibration, so is Grpd((S 1//G) ,(p ρ 2) )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

ϕ :(S 1//G) (S 1//G) \phi_\bullet \colon (S_1//G)_\bullet \longrightarrow (S_1//G)_\bullet

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

ϕ :(s g ρ(s)(g))(ϕ(s) g ϕ(ρ(s)(g)) =ρ(ϕ(s))(g)). \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. 1 is equivalent to the set SS, regarded as a groupoid with only identity morphisms, hence we have a homotopy fiber sequence of the form

S S//G p ρ BG. \array{ S &\longrightarrow& S//G \\ && \downarrow^{\mathrlap{p_\rho}} \\ && \mathbf{B}G } \,.
Proof

In the presentation (S//G) (S//G)_\bullet of def. 1, p ρp_\rho is an isofibration, prop. 1. Hence the homotopy fibers of p ρp_\rho are equivalent to the ordinary fibers of (p ρ) (p_\rho)_\bullet computed in the 1-category of 1-groupoids. Since (p ρ) (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) (S//G)_\bullet consisting of only the identity morphismss, hence is the set SS 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

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

such that SS is equivalent to a set SS, define a GG-action on this set as follows.

Consider the homotopy fiber product

S×ESS S \underset{E}{\times} S \stackrel{\overset{}{\longrightarrow}}{\underset{}{\longrightarrow}} S

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

S×ES S i S i E p * BGS×G p 1 S G * * BG \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×ESS×G S \underset{E}{\times} S \simeq S \times G

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

S×Gρp 1S. S \times G \stackrel{\overset{p_1}{\longrightarrow}}{\underset{\rho}{\longrightarrow}} S \,.
Remark

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

E (S×ESp 2p 1S) 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. 2, by

E (S×Gρp 1S). E_\bullet \simeq (S \times G \stackrel{\overset{p_1}{\longrightarrow}}{\underset{\rho}{\longrightarrow}} S) \,.

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

Proposition

The morphism ρ\rho constructed in def. 2 is a GG-action in that it satisfies the action propery, which says that the diagram (of sets)

S×G×G (id,()()) S×G (ρ,id) ρ S×G ρ S \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 GG a discrete group, there is an equivalence of categories

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

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

This equivalence takes an action to its action groupoid.

Proof

By remark 4 the construction of action groupoids is essentially surjective. By prop. 2 it is fully faithful.

Examples of actions

One remarkable consequence of prop. 5 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 GG and two GG-actions ρ 1\rho_1 and ρ 2\rho_2 on sets S 1S_1 and S 2S_2, respectively, then the function set [S 1,S 2][S_1, S_2] is naturally equipped with the conjugation action

Ad:[S 1,S 2]×G[S 1,S 2] Ad \;\colon \; [S_1, S_2] \times G \longrightarrow [S_1,S_2]

which takes ((S 1fS 2),g)((S_1 \stackrel{f}{\to} S_2), g) to

ρ 2()(g)fρ 1()(g 1):S 1ρ 1()(g 1)S 1fS 2ρ 2()(g)S 2. \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. 3 is the internal hom in the category of actions.

Proof

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

ϕ:S 3[S 1,S 2] \phi \;\colon\; S_3 \longrightarrow [S_1,S_2]

which intertwine the GG-action on S 3S_3 with the conjugation action on [S 1,S 2][S_1,S_2] are in natural bijection with functions

ϕ˜:S 3×S 1S 2 \tilde \phi \;\colon\; S_3 \times S_1 \longrightarrow S_2

which intertwine the diagonal action on the Cartesian product S 3×S 1S_3 \times S_1 with the action on S 2S_2.

The condition on ϕ\phi means that for all gGg\in G and s 3S 3s_3 \in S_3 it sends

ϕ:ρ 3(s 3)(g)(s 1ρ 2(ϕ(s 3)(ρ 1(s 1)(g 1)))(g)). \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 ϕ˜\tilde \phi of two variables which sends

ϕ˜:(ρ 3(s 3)(g),s 1)ρ 2(ϕ(s 3)(ρ 1(s 1)(g 1)))(g). \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 1s_1 with ρ 1(s 1)(g)\rho_1(s_1)(g). After this substitution the above becomes

ϕ˜:(ρ 3(s 3)(g),ρ 1(s 1)(g))ρ 2(ϕ(s 3)(s 1))(g). \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 ϕ˜\tilde \phi. Conversely, given ϕ˜\tilde \phi satisfying this for all values of the variables, then running the argument backwards shows that its hom-adjunct ϕ\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][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 GG-actions into the set underlying the internal hom

GAct(ρ 1,ρ 2)[ρ 1,ρ 2]. G Act(\rho_1,\rho_2)\hookrightarrow [\rho_1,\rho_2] \,.
Example

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

Then the conjugation action, def. 3, on [X,Y][X,Y] is the action by precomposition with automorphisms of XX.

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 GG over a smooth manifold XX are equivalently the homotopy fibers of morphisms of smooth groupoids (smooth stacks) of the form

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

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

V//G p ρ X g BG \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 PXP \to X via the action ρ\rho.

Proposition

For GG a smooth group (e.g. a Lie group), XX a smooth manifold, PXP \to X a smooth GG-principal bundle over XX and ρ\rho a smooth action of GG on some smooth manifold VV, then the associated VV-fiber bundle P× GVXP \times_G V\to X is equivalently (regarded as a smooth groupoid) the homotopy pullback of the action groupoid-projection p ρ:V//GBGp_\rho \colon V//G \to \mathbf{B}G along a morphism g:XmathgbfBGg \colon X\to\mathgbf{B}G which modulates PP

P× GV V//G p ρ X g BG. \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 gg of smooth groupoids is presented by a morphism of pre-smooth groupoids after choosing an open cover {U iX}\{U_i \to X\} over wich PP trivialize and choosing a trivialization, by the zig-zag

C({U i}) g (BG) lwe X \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. 1 the morphism (p ρ) (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 ρ) (p_\rho)_\bullet along this g g_\bullet

C({U i}) ×(BG) (V//G) (V//G) (p ρ) C({U i}) g (BG) lwe X. \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)(x,j)(x,i)\to (x,j) in the Cech groupoid as well as a morphism sgρ(s)(g)s \stackrel{g}{\to} \rho(s)(g) in the action groupoid, such that the group label gg of the latter equals the cocycle g ij(x)g_{i j}(x) of the cocycle on the former. Schematically:

C({U i}) ×(BG) (V//G) ={((x,i),s)g ij(x)((x,j),ρ(s)(g))}. 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:

(iU i×V)/ . \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 GG a discrete group, ρ\rho a GG-action on some set SS, then the set of invariants of that action is equivalent to the groupoid of sections of the action groupoid projection of prop. 1, corresponding to the action via prop. 5.

Proof

The sections in question are diagrams in Grpd of the form

BG σ S//G id p ϕ BG, \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 /BG(id BG,p ρ)Grpd Grpd_{/\mathbf{B}G}(id_{\mathbf{B}G}, p_\rho) \in Grpd

in the slice of Grpd over BG\mathbf{B}G. As in the proof of prop. 2, with the fibrant presentation (p ρ) (p_\rho)_\bullet of prop. 1, this is equivalently given by strictly commuting diagrams of the form

(BG) σ (S//G) id = (p ϕ) (BG) . \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 σ\sigma now are manifestly functors that are the identiy on the group labels of the morphisms

σ :(* g *)(σ(*) g σ(*) =ρ(σ(*)(g))). \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 σ(*)S\sigma(\ast) \in S which are fixed by the GG-action ρ\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× GVXP \times_G V\to X modulated, as in prop. 8, by a morphism of smooth groupoids of the form g:XBGg \colon X \longrightarrow \mathbf{B}G, then its set of sections is equivalently the groupoid of diagrams

X σ S//G g p ϕ BG, \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

Γ X(P× GV)Grpd /BG(g,p ρ). \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. 8.

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 GG and an action ρcolobG×VV\rho \colob G \times V \longrightarrow V of the group on some set VV, then for xVx \in V any element, the stabilizer group

Stab x(V)G Stab_x(V) \hookrightarrow G

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

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

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

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

for the action groupoid homotopy fiber sequence that corresponds to ρ\rho via prop. 5.

Proposition

The stabilizer group of xx under GG acting on VV is equivalently the looping of the action groupoid at the point i(x)i(x):

Stab G(x)Ω i(x)(V//G) Stab_G(x) \simeq \Omega_{i(x)} (V//G)
Proof

Since V//GV//G is a 1-groupoid, the loop space object Ω i(x)(V//G)\Omega_{i(x)} (V//G) is simply the automorphism group of xx regarded as an object in the action groupoid, hence the group of morphisms in (V//G) (V//G)_\bullet from xx to xx. Comparison with def. 1 shows that these morphisms are precisely those labeled by elements gGg \in G for which ρ(g)(x)=x\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 GG and a Kan complex V V_\bullet, then an infinity-representation or infinity-action of GG on VV is another Kan complex, to be denoted (V//G) (V//G)_\bullet, equipped with a simplicial map (p ρ):(V//G) N(BG) (p_\rho) \colon (V//G)_\bullet \longrightarrow N(\mathbf{B}G)_\bullet to the nerve of (BG )(\mathbf{B}G_\bullet), such that the homotopy fiber of that map is weakly homotopy equivalent to V V_\bullet.

Examples of \infty-Representations

Given an abelian group AA and nn \in \mathbb{N}, write (B nA) (\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 AA in degree nn.

Then for GG a discrete group, the mapping complex

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

is the infinity-groupoid whose objects are the degree-nn group cocycles on GG with coefficients in AA (regarded as a GG-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 BG\mathbf{B}G, which is also known as the automorphism 2-group of GG. Restricting this to pointed automorphisms is is the 1-group Aut Grp(G)Aut_{Grp}(G) of invertible group homomorphisms of GG.

Semantic Layer

We now consider any (∞,1)-topos H\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 (*BG)H */(\ast\to\mathbf{B}G)\in \mathbf{H}^{\ast/}, the group itself being the loop space object

GΩBG 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. 5 has an immediate generalization to ∞-groups in any (∞,1)-topos

Definition

Given an ∞-group GHG \in \mathbf{H} and any object VHV \in \mathbf{H}, then an ∞-action ρ\rho of GG on VV is a homotopy fiber sequence in H\mathbf{H} of the form

V VG p ρ BG. \array{ V &\to& V\sslash G \\ && \downarrow^{\mathrlap{p_\rho}} \\ && \mathbf{B}G } \,.

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

Associated \infty-bundles

E VG pb X˜ BG X \array{ E &\to& V\sslash G \\ \downarrow &pb& \downarrow \\ \tilde X &\to& \mathbf{B}G \\ \downarrow^{\mathrlap{\simeq}} \\ X }
X σ VG BG \array{ X &&\stackrel{\sigma}{\to}&& V \sslash G \\ & \searrow &\swArrow_{\simeq}& \swarrow \\ && \mathbf{B}G }

Stabilizer \infty-Groups

We had seen above in prop. 11 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. 5.

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

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

Given a global element of VV

x:*X x \colon \ast \to X

then the stabilizer \infty-group Stab ρ(x)Stab_\rho(x) of the GG-action at xx is the loop space object

Stab ρ(x)Ω i(x)(X/G). Stab_\rho(x) \coloneqq \Omega_{i(x)} (X/G) \,.
Remark

Equivalently, def. 6, gives the loop space object of the 1-image BStab ρ(x)\mathbf{B}Stab_\rho(x) of the morphism

*xXX/G. \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 *BStab ρ(x)X/G\ast \to \mathbf{B}Stab_\rho(x) \hookrightarrow X/G which combines with the homotopy fiber sequence of prop. 5 to a diagram of the form

* x X X/G epi mono BStab ρ(x) = BStab ρ(x) BG. \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 ρ(x)G. 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 ρ(x)Stab_\rho(x) is not a sub-group of GG 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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Revised on May 20, 2015 07:25:06 by Urs Schreiber (195.113.30.252)