nLab
infinity-action

Contents

Idea

The notion of \infty-action is the notion of action (module/representation) in homotopy theory/(∞,1)-category theory, from algebra to higher algebra.

Notably a monoid object in an (∞,1)-category AA may act on another object NN by a morphism ANNA \otimes N \to N which satisfies an action property up to coherent higher homotopy.

If the \infty-action is suitably linear in some sense, this is also referred to as ∞-representation.

Definition

We discuss the actions of ∞-groups in an (∞,1)-topos. (For groupoid ∞-actions see there.)

Let H\mathbf{H} be an (∞,1)-topos.

Let GGrp(H)G \in Grp(\mathbf{H}) be an group object in an (∞,1)-category in H\mathbf{H}, hence a homotopy-simplicial object on H\mathbf{H} of the form

(G×GG*) \left( \cdots \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}} G \times G \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} G \stackrel{\longrightarrow}{\longrightarrow} * \right)

satisfying the groupoidal Segal conditions.

hence an ∞-group.

Definition

An action (or \infty-action, for emphasis) of GG on an object VHV \in \mathbf{H} is a groupoid object in an (∞,1)-category which is equivalent to one of the form

(V×G×GV×Gp 1ρV) \left( \cdots \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}} V \times G \times G \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} V \times G \stackrel{\overset{\rho}{\longrightarrow}}{\underset{p_1}{\longrightarrow}} V \right)

such that the projection maps

V×G×G V×G p 1ρ V G×G G * \array{ \cdots &\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}}& V \times G \times G &\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}& V \times G &\stackrel{\overset{\rho}{\longrightarrow}}{\underset{p_1}{\longrightarrow}}& V \\ && \downarrow && \downarrow && \downarrow \\ \cdots &\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}}& G \times G &\stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}}& G &\stackrel{\overset{}{\longrightarrow}}{\underset{}{\longrightarrow}}& * }

constitute a morphism of groupoid objects VG*GV\sslash G \to *\sslash G.

The (∞,1)-category of such actions is the slice of groupoid objects over *G*\sslash G on these objects.

There is an equivalent formulation which does not invoke the notion of groupoid object in an (∞,1)-category explicitly. This is based on the fundamental fact, discussed at ∞-group, that delooping constitutes an equivalence of (∞,1)-categories

B:Grp(H)H 1 */. \mathbf{B} : Grp(\mathbf{H}) \to \mathbf{H}^{*/}_{\geq 1} \,.

form group objects in an (∞,1)-category to the (∞,1)-category of connected pointed objects in H\mathbf{H}.

Proposition

Every \infty-action ρ:V×GV\rho : V \times G \to V has a classifying morphism c ρ:VGBG\mathbf{c}_\rho : V \sslash G \to \mathbf{B}G in that there is a fiber sequence

V VG ρ¯ BG \array{ V \\ \downarrow \\ V \sslash G &\stackrel{\overline{\rho}}{\to}& \mathbf{B}G }

such that ρ\rho is the GG-action on VV regarded as the corresponding GG-principal ∞-bundle modulated by ρ¯\overline{\rho}.

This allows to characterize \infty-actions in the following convenient way. See (NSS) for a detailed discussion.

Definition

For VHV \in \mathbf{H} an object, a GG-\infty-action ρ\rho on VV is a fiber sequence in H\mathbf{H} of the form

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

The (∞,1)-category of GG-actions in H\mathbf{H} is the slice (∞,1)-topos of H\mathbf{H} over BG\mathbf{B}G:

Act H(G)H /BG. Act_{\mathbf{H}}(G) \coloneqq \mathbf{H}_{/\mathbf{B}G} \,.
Remark

A ρAct H(G)\rho \in Act_{\mathbf{H}}(G) corresponds to a morphism denoted ρ¯:VGBG\overline{\rho} : V\sslash G \to \mathbf{B}G in H\mathbf{H} hence to an object ρ¯H /BG\overline{\rho} \in \mathbf{H}_{/\mathbf{B}G}.

A morphism ϕ:ρ 1ρ 2\phi : \rho_1 \to \rho_2 in Act H(G)Act_{\mathbf{H}}(G) corresponds to a diagram

V 1G V 2G ρ 1¯ ρ 2¯ BG \array{ V_1 \sslash G &&\stackrel{}{\to}&& V_2 \sslash G \\ & {}_{\mathllap{\overline{\rho_1}}}\searrow && \swarrow_{\mathrlap{\overline{\rho_2}}} \\ && \mathbf{B}G }

in H\mathbf{H}.

Remark

The bundle ρ¯\overline{\rho} in def. 2 is the universal ρ\rho-associated VV-fiber ∞-bundle.

Remark

In the form of def. 2 \infty-actions have a simple formulation in the internal language of homotopy type theory: a GG-action on VV is simply a dependent type over BG\mathbf{B}G with fiber VV:

*:BGV(*):Type. * : \mathbf{B}G \vdash V(*) : Type \,.

Notions in higher representation theory

We discuss some basic representation theoretic notions of \infty-actions.

In summary, for c:BGV(c):Type\mathbf{c} : \mathbf{B}G \vdash V(\mathbf{c}) : Type an action of GG on VV, we have

And for V 1,V 2V_1, V_2 two actions we have

Invariants

Definition

The invariants (homotopy fixed points) of a GG-\infty-action ρ\rho are the sections of the morphism VGBGV \sslash G \to \mathbf{B}G,

Invariants(V)= BG*(VGBG), Invariants(V) = \prod_{\mathbf{B}G \to *} (V \sslash G \to \mathbf{B}G) \,,

where BG*:H /BGH\prod_{\mathbf{B}G \to *} : \mathbf{H}_{/\mathbf{B}G} \to \mathbf{H} is the direct image of the base change geometric morphism.

In homotopy type theory syntax for

c:BGV(c):Type \mathbf{c} : \mathbf{B}G \vdash V(\mathbf{c}) : Type

an action as in remark 3, its type of invariants is the dependent product

c:BGV(c):Type. \vdash \prod_{\mathbf{c} : \mathbf{B}G} V(\mathbf{c}) : Type \,.
Remark

This is the internal limit in H\mathbf{H} of the internal diagram

ρ:BGType. \rho \colon \mathbf{B}G \to Type \,.

See at internal limit – Examples – Homotopy Invariants.

Coinvariants / Quotients

From def. 2 we read off:

Definition

The quotient of a GG-action

c:BGV(c):Type \mathbf{c} : \mathbf{B}G \vdash V(\mathbf{c}) : Type

is the dependent sum

c:BGV(c):Type. \vdash \sum_{\mathbf{c} : \mathbf{B}G} V(\mathbf{c}) : Type \,.
Remark

This is the internal colimit in H\mathbf{H} of the internal diagram

ρ:BGType. \rho \colon \mathbf{B}G \to Type \,.

See at internal limit – Examples – Homotopy Coinvariants.

Conjugation actions

Remark

By def. 2, and basic facts disussed at slice (∞,1)-topos, the (∞,1)-category Act H(G)Act_{\mathbf{H}}(G) is an (∞,1)-topos and in particular is a cartesian closed (∞,1)-category.

We describe here aspects of the cartesian product and internal hom of \infty-actions given this way. The following statements are essentially immediate consequences of basic homotopy type theory.

Proposition

For (V 1,ρ 1),(V 2,ρ 2)Act(G)(V_1, \rho_1), (V_2, \rho_2) \in Act(G) their cartesian product is a GG-action on the product of V 1V_1 with V 2V_2 in H\mathbf{H}.

Proof

Let

V i V iG ρ¯ i BG \array{ V_i &\to& V_i \sslash G \\ && \downarrow^{\bar \rho_i} \\ && \mathbf{B}G }

be the principal ∞-bundles exhibiting the two actions.

Along the lines of the discussion at locally cartesian closed category we find that (V 1,ρ 1)×(V 2,ρ 2)Act(G)(V_1, \rho_1) \times (V_2, \rho_2) \in Act(G) is given in H\mathbf{H} by the (∞,1)-pullback

BGρ¯ 1×ρ¯ 2V 1G× BGV 2G \sum_{\mathbf{B}G} \bar \rho_1 \times \bar \rho_2 \simeq V_1\sslash G \times_{\mathbf{B}G} V_2 \sslash G

in H\mathbf{H}, with the product action being exhibited by the principal ∞-bundle

V 1×V 2 V 1G× BGV 2G ρ 1×ρ 2¯ BG. \array{ V_1 \times V_2 &\to& V_1\sslash G \times_{\mathbf{B}G} V_2 \sslash G \\ && \downarrow^{\mathrlap{\overline{ \rho_1 \times \rho_2 }}} \\ && \mathbf{B}G } \,.

Here the homotopy fiber on the left is identified as V 1×V 2V_1 \times V_2 by using that (∞,1)-limits commute over each other.

Proposition

For ρ 1,ρ 2Act(G)\rho_1, \rho_2 \in Act(G) their internal hom [ρ 1,ρ 2]Act H(G)[\rho_1, \rho_2] \in Act_{\mathbf{H}}(G) is a GG-action on the internal hom [V 1,V 2]H[V_1, V_2] \in \mathbf{H}.

Proof

Taking fibers

pt BG *:H /BGH pt_{\mathbf{B}G}^* : \mathbf{H}_{/\mathbf{B}G} \to \mathbf{H}

is the inverse image of an etale geometric morphism, hence is a cartesian closed functor (see the Examples there for details). Therefore it preserves exponential objects:

pt BG *[ρ¯ 1,ρ¯ 2] [pt BG *ρ¯ 1,pt BG *ρ¯ 2] [V 1,V 2]. \begin{aligned} pt_{\mathbf{B}G}^* [\bar \rho_1, \bar \rho_2] & \simeq [pt_{\mathbf{B}G}^* \bar \rho_1, pt_{\mathbf{B}G}^* \bar \rho_2] \\ & \simeq [V_1, V_2] \end{aligned} \,.
Remark

The above internal-hom action

[V 1,V 2] V 1G× BGV 2G [ρ 1,ρ 2]¯ BG \array{ [V_1,V_2] &\to& V_1 \sslash G \times_{\mathbf{B}G} V_2 \sslash G \\ && \downarrow^{\mathrlap{\overline{[\rho_1,\rho_2]}}} \\ && \mathbf{B}G }

encodes the conjugation action of GG on [V 1,V 2][V_1, V_2] by pre- and post-composition of functions V 1V 2V_1 \to V_2 with the GG-action on V 1V_1 and on V 2V_2, respectively.

See also at Conjugation actions below.

Internal object of homomorphisms

Remark

The invariant, def. 3 of the conjugation action, prop. 3 are the action homomorphisms. (See also at Examples - Conjugation actions.)

Therefore

Definition

For ρ¯ i:V iGBG\bar \rho_i : V_i \sslash G \to \mathbf{B}G two GG-actions, the object of homomorphisms is

BG*[ρ¯ 1,ρ¯ 2]H. \prod_{\mathbf{B}G \to *}[\bar \rho_1, \bar \rho_2] \in \mathbf{H} \,.

In the syntax of homotopy type theory

c:BGV 1(c)V 2(c):Type. \vdash \prod_{\mathbf{c} : \mathbf{B}G} V_1(\mathbf{c}) \to V_2(\mathbf{c}) : Type \,.

Stabilizer groups

See at stabilizer group.

Linearization

We discuss linearization of \infty-actions using the axioms of differential cohesion.

Let 0:*V0 \colon \ast \to V be a pointed object.

Let GG be an \infty-group acting on VV

V V/G BG \array{ V &\longrightarrow& V/G \\ && \downarrow \\ && \mathbf{B}G }

such that this action preserves the point of VV, i.e. such that the point is an invariant of the action. This means equivalently that there is a lift as given by the diagonal morphism in

* 0 V V/G * BG \array{ \ast &\stackrel{0}{\longrightarrow}& V &\longrightarrow & V/G \\ & \searrow& & \nearrow& \downarrow \\ && \ast &\longrightarrow& \mathbf{B}G }

which in turn means that the action factors through an action of the stabilizer group Stab G(0)Stab_G(0)

* BStab G(0) BG V/G BG \array{ \ast &\longrightarrow& \mathbf{B}Stab_G(0) \\ \downarrow &\nearrow& \downarrow & \searrow \\ \mathbf{B}G &\longrightarrow& V/G &\longrightarrow& \mathbf{B}G }

(using that the left morphism is a 1-epimorphism and the right morphism a 1-monomorphism).

It follows by the pasting law the top squares in the following diagram is a homotopy pullback

* */G 0 V V/G * BG \array{ \ast &\longrightarrow& \ast/G \\ {}^{\mathllap{0}}\downarrow && \downarrow \\ V &\longrightarrow& V/G \\ \downarrow && \downarrow \\ \ast &\longrightarrow& \mathbf{B}G }

exhibiting that the GG-action on VV restricts to the trivial action on the point 00 of VV.

Now let inf\int_{inf} denote the infinitesimal shape modality. Since it preserves the top homotopy pullback, it follows that applying the orthogonal factorization system ( inf\int_{inf}-equivalences, formally etale morphisms) to the top vertical morphisms produces a pasting diagram of homotopy pullbacks of the form

* */G 𝔻 0 V 𝔻 0 V/G V V/G * BG \array{ \ast &\longrightarrow& \ast/G \\ \downarrow && \downarrow \\ \mathbb{D}^V_0 &\longrightarrow& \mathbb{D}^V_0/G \\ \downarrow && \downarrow \\ \downarrow && \downarrow \\ V &\longrightarrow& V/G \\ \downarrow && \downarrow \\ \ast &\longrightarrow& \mathbf{B}G }

where 𝔻 0 V\mathbb{D}^V_0 is the infinitesimal disk around 00 in VV.

Here the cartesian subdiagram

𝔻 0 V 𝔻 0 V/G * BG \array{ \mathbb{D}^V_0 &\longrightarrow& \mathbb{D}^V_0/G \\ \downarrow && \downarrow \\ \ast &\longrightarrow& \mathbf{B}G }

hence exhibits a GG-action on 𝔻 0 V\mathbb{D}^V_0.

Any GG-action on an infinitesimal disk is a linear action, given by a homomorphism GGL(V)Aut(𝔻 0 V)G \to GL(V) \coloneqq \mathbf{Aut}(\mathbb{D}^V_0) to the automorphism infinity-group of the infinitesimal disk, the general linear group of the tangent space of VV at 0.

Examples

Discrete group actions on sets

As the simplest special case, we discuss how the traditional concept of discrete groups acting on a sets (“permutation representations”) is recoverd from the above general abstract concepts.

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. 6, 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. 4

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. 4 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. 6, p ρp_\rho is an isofibration, prop. 4. 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. 7, 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. 7 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 7 the construction of action groupoids is essentially surjective. By prop. 5 it is fully faithful.

\infty-group actions in an \infty-topos

Let H\mathbf{H} be an (∞,1)-topos and let GGrp(H)G \in Grp(\mathbf{H}) be an ∞-group in H\mathbf{H}.

The following lists some fundamental classes of examples of \infty-actions of GG, and of other canonical \infty-groups. By the discussion above these actions may be given by the classifying morphisms.

Trivial action

Consider the étale geometric morphism

Act H(G)H /BGp *()×BGH. Act_{\mathbf{H}}(G) \coloneqq \mathbf{H}_{/\mathbf{B}G} \stackrel{\overset{p^* \coloneqq (-) \times \mathbf{B}G}{\leftarrow}}{\underset{}{\to}} \mathbf{H} \,.
Definition

For VHV \in \mathbf{H} any object, the trivial action of GG on VV is p *VAct H(G)p^* V \in Act_{\mathbf{H}}(G), exhibited by the split fiber sequence

V V×BG BG. \array{ V &\to& V \times \mathbf{B}G \\ && \downarrow \\ && \mathbf{B}G } \,.

Fundamental action

The right \infty-action of GG on itself is given by the fiber sequence

G * BG \array{ G \\ \downarrow \\ * &\to& \mathbf{B}G }

which exhibits BG\mathbf{B}G as the delooping of GG.

GG*. G \sslash G \simeq * \,.

Adjoint action

The fiber sequence

G BG ev * BG \array{ G \\ \downarrow \\ \mathcal{L} \mathbf{B}G &\stackrel{ev_*}{\to}& \mathbf{B}G }

given by the free loop space object BG\mathcal{L}\mathbf{B}G exhibits the higher adjoint action of GG on itself:

G AdGBG. G \sslash_{Ad}G \simeq \mathcal{L}\mathbf{B}G \,.

Automorphism action

Definition

For VHV \in \mathbf{H} any object, there is a canonical action of the internal automorphism infinity-group Aut(V)\mathbf{Aut}(V):

V VAut(V) BAut(V) \array{ V \\ \downarrow \\ V \sslash \mathbf{Aut}(V) &\to& \mathbf{B} \mathbf{Aut}(V) }

Conjugation actions

We discuss the simple case of the cartesian closed category of GG-sets (G-permutation representations) for GG an ordinary discrete group as a simple illustration of the internal hom of \infty-actions, prop. 3.

This example spells out everything completely in components:

Example

Let H=\mathbf{H} = ∞Grpd, let GGrp(Grpd)G \in Grp(\infty Grpd) be an ordinary discrete group and let V,Σ,XV, \Sigma, X be sets equipped with GG-action (permutation representations).

In this case [Σ,X][\Sigma,X] is simply the set of functions f:ΣXf : \Sigma \to X of sets. Its GG-action as the internal hom of GG-actions given, for every gGg \in G and σΣ\sigma \in \Sigma, by

g(f)(σ)=g(f(g 1(σ))), g(f)(\sigma) = g(f(g^{-1}(\sigma))) \,,

(where we write generically g()g(-) for the given action on the set specified implicitly by the type of the argument).

Hence a morphism of GG-actions

ϕ:V[Σ,X] \phi : V \to [\Sigma,X]

is a function ϕ\phi of the underlying sets such that for all VVV \in V, gGg \in G and all σΣ\sigma \in \Sigma we have

(1)ϕ(g(v))(σ)=g(ϕ(v)(g 1(σ)). \phi(g(v))(\sigma) = g(\phi(v)(g^{-1}(\sigma)) \,.

On the other hand, a morphism of actions

ψ:V×ΣX \psi : V \times \Sigma \to X

is a function of the underlying sets, such that for all these terms we have

ψ(g(v),g(σ))=g(ψ(v,σ)) \psi(g(v), g(\sigma)) = g(\psi(v,\sigma))

which is equivalent to

(2)ψ(g(v),σ)=g(ψ(v,g 1(σ))). \psi(g(v), \sigma) = g(\psi(v,g^{-1}(\sigma))) \,.

Comparison of (1) and (2) shows that the identification

ψ(v,σ)ϕ(v)(σ) \psi(v,\sigma) \coloneqq \phi(v)(\sigma)

establishes a natural equivalence (a natural bijection of sets in this case)

Act H(G)(V,[Σ,X])Act H(G)(V×Σ,X), Act_{\mathbf{H}}(G)(V, [\Sigma,X]) \simeq Act_{\mathbf{H}}(G)(V \times \Sigma, X) \,,

showing how [Σ,X][\Sigma,X] is indeed the internal hom of GG-actions.

Remark

Generally, for GG a discrete ∞-group we have an equivalence of (∞,1)-categories

Grpd /BGFunc(BG,Grpd) \infty Grpd_{/\mathbf{B}G} \simeq \infty Func(\mathbf{B}G, \infty Grpd)

(by the (∞,1)-Grothendieck construction), and hence

Act Grpd(G)Func(BG,Grpd) Act_{\infty Grpd}(G) \simeq \infty Func(\mathbf{B}G, \infty Grpd)

is the (∞,1)-category of ∞-permutation representations.

General covariance

Let XHX \in \mathbf{H} be a moduli infinity-stack for field in a gauge theory or sigma-model. Let ΣH\Sigma \in \mathbf{H} be the corresponding spacetime or worldvolume, respectively.

We have the automorphism action, def. 9

Σ ΣAut(Σ) BAut(Σ). \array{ \Sigma &\to& \Sigma \sslash \mathbf{Aut}(\Sigma) \\ && \downarrow \\ && \mathbf{B} \mathbf{Aut}(\Sigma) } \,.

The slice H /Aut(Σ)=Act H(Aut(Σ))\mathbf{H}_{/\mathbf{Aut}(\Sigma)} = Act_{\mathbf{H}}(\mathbf{Aut}(\Sigma)) is the context of types which are generally covariant over Σ\Sigma.

On XX consider the trivial Aut(Σ)\mathbf{Aut}(\Sigma)-action, def. 8. Then the internal-hom action of prop. 3

[Σ,X]Aut(Σ)[ΣAut(Σ),X×BAut(Σ)] BAut(Σ) [\Sigma, X]\sslash \mathbf{Aut}(\Sigma) \simeq [\Sigma \sslash \mathbf{Aut}(\Sigma), X \times \mathbf{B}\mathbf{Aut}(\Sigma)]_{\mathbf{B}\mathbf{Aut}(\Sigma)}

is the configuration space of fields on Σ\Sigma modulo automorphisms (diffeomorphisms, in smooth cohesion) of Σ\Sigma. This is the configuration space of “generally covariant” field theory on Σ\Sigma.

Semidirect product groups

Let G,AGrp(H)G, A \in Grp(\mathbf{H}) be 0-truncated group objects and let ρ\rho be an action of GG on AA by group homomorphisms. This is equivalently an action of GG on BA\mathbf{B}A, hence a fiber sequence

BA B(GA) BG. \array{ \mathbf{B}A &\to& \mathbf{B} (G \ltimes A) \\ && \downarrow \\ && \mathbf{B}G } \,.

The corresponding action groupoid (BA)GB(GA)(\mathbf{B}A)\sslash G \simeq \mathbf{B}( G \ltimes A) is the delooping of the corresponding semidirect product group.

GG-Modules

Definition

For GGrp(H)G \in Grp(\mathbf{H}) the \infty-category of GG-modules is

Stab(H /BG)Stab(GAct), Stab( \mathbf{H}_{/\mathbf{B}G}) \simeq Stab(G Act) \,,

the stabilization of the \infty-category of GG-actions.

Example

For GG and AA 0-truncated groups, AA an abelian group with GG-module structure, the semidirect product group GAG \ltimes A from above exhibits AA as a GG-module in the sense of def. 10.

Actions in a slice

Consider an object BHB \in \mathbf{H} and an object

LH /B L \in \mathbf{H}_{/B}

in the slice. By the discussion of conjugation actions above, the automorphism ∞-group of LL as an object in H\mathbf{H} is the dependent product over the automorphism ∞-group Aut H(L)H /B\mathbf{Aut}_{\mathbf{H}}(L)\in \mathbf{H}_{/B} in the slice.

Aut H(L)BAut(L)Grp(H). \mathbf{Aut}_{\mathbf{H}}(L) \coloneqq \underset{B}{\prod} \mathbf{Aut}(L) \in \mathrm{Grp}(\mathbf{H}) \,.

By adjunction there is a canonical morphism from the re-pullback of this to the slice automorphism group

ϵ:B *BAut H(L)BAut(L). \epsilon \colon B^\ast \mathbf{B}\mathbf{Aut}_{\mathbf{H}}(L) \longrightarrow \mathbf{B} \mathbf{Aut}(L) \,.

Hence the canonical Aut(L)\mathbf{Aut}(L)-action on LL in the slice pulls back to give an action of B *Aut H(L)B^\ast \mathbf{Aut}_{\mathbf{H}}(L) on LL:

L L//(B *Aut H(L)) L//Aut(L) * BB *Aut H(L) ϵ BAut(L) \array{ L &\longrightarrow& L//(B^\ast\mathbf{Aut}_{\mathbf{H}}(L)) &\longrightarrow& L//\mathbf{Aut}(L) \\ \downarrow && \downarrow && \downarrow \\ \ast &\longrightarrow& \mathbf{B}B^\ast \mathbf{Aut}_{\mathbf{H}}(L) &\stackrel{\epsilon}{\longrightarrow}& \mathbf{B} \mathbf{Aut}(L) }
Proposition

Underlying the B *Aut H(L)B^\ast\mathbf{Aut}_{\mathbf{H}}(L)-action on LL is an Aut H(L)\mathbf{Aut}_{\mathbf{H}}(L)-action on

XBL X \coloneqq \underset{B}{\sum} L

and

B(L//B *Aut H(L))X//Aut H(L) \underset{B}{\sum} \left(L//B^\ast\mathbf{Aut}_{\mathbf{H}}(L)\right) \;\simeq\; X//\mathbf{Aut}_{\mathbf{H}}(L)
Proof

Applying B\underset{B}{\sum} to the Cartesian diagram that defines the \infty-action on LL

L L//Aut H(L) * BB *Aut H(L) \array{ L &\longrightarrow& L//\mathbf{Aut}_{\mathbf{H}}(L) \\ \downarrow && \downarrow \\ \ast &\longrightarrow& \mathbf{B}B^\ast \mathbf{Aut}_{\mathbf{H}}(L) }

yields

X X(L//Aut H(L)) B BB *BAut H(L) \array{ X &\longrightarrow& \underset{X}{\sum} \left( L//\mathbf{Aut}_{\mathbf{H}}(L) \right) \\ \downarrow && \downarrow \\ B &\longrightarrow& \underset{B}{\sum} B^\ast \mathbf{B} \mathbf{Aut}_{\mathbf{H}}(L) }

which is still Cartesian, by this proposition. Use that the bottom left object here is equivalently BBB *(*)B \simeq \underset{B}{\sum} B^\ast (\ast) and form the pasting with the naturality square of the (BB *)(\underset{B}{\sum}\dashv B^\ast)-counit.

X B(L//Aut H(L)) BB ** BB *BAut H(L) * BAut H(L). \array{ X &\longrightarrow& \underset{B}{\sum} \left(L//\mathbf{Aut}_{\mathbf{H}}(L)\right) \\ \downarrow && \downarrow \\ \underset{B}{\sum}B^\ast \ast &\longrightarrow& \underset{B}{\sum}B^\ast \mathbf{B}\mathbf{Aut}_{\mathbf{H}}(L) \\ \downarrow && \downarrow \\ \ast &\longrightarrow& \mathbf{B}\mathbf{Aut}_{\mathbf{H}}(L) } \,.

By this proposition also this naturality square is Cartesian. Hence by the pasting law the total rectangle is Cartesian. This exhibits the Aut H(L)\mathbf{Aut}_{\mathbf{H}}(L)-action on X=BLX = \underset{B}{\sum} L.

Remark

Stated more intuitively, prop. 9 says that sliced automorphisms of the form

Aut H(L)={X X L L B} \mathbf{Aut}_{\mathbf{H}}(L) = \left\{ \array{ X & & \stackrel{\simeq}{\longrightarrow} & & X \\ & {}_{\mathllap{L}}\searrow &\swArrow_{\simeq}& \swarrow_{\mathrlap{L}} \\ && B } \right\}

act on XX by the evident restriction to the horizontal equivalences,

{X X} \left\{ \array{ X & & \stackrel{\simeq}{\longrightarrow} & & X } \right\}

and that forming the homotopy quotient of this action on LL makes LL descent to the homotopy quotient of XX by this action to yield

X//Aut H(L) L//Aut H(L) B. \array{ X // \mathbf{Aut}_{\mathbf{H}}(L) \\ \downarrow^{\mathrlap{L//\mathbf{Aut}_{\mathbf{H}}(L)}} \\ B } \,.

(For instance if here BB is a moduli stack for some prequantum n-bundles, then this says that the quantomorphism n-group acting on this gives higher and pre-quantized “symplectic reduction” of these bundles to the quotient space.)

Co-Discretization of Actions

Let H\mathbf{H} be a local (∞,1)-topos (for instance a cohesive (∞,1)-topos) and write \sharp for its sharp modality. Write n\sharp_n for the n-image of itd unit.

Proposition

Given an ∞-group GG in H\mathbf{H} and a GG-action, def. 2, on some XX, then nG\sharp_n G is itself canonically an \infty-group equipped with a canonically induced action on nX\sharp_n X such that the projection X nXX \to \sharp_n X carries the structure of a homomorphism of GG-actions.

We indicate two proofs, the first non-elementary (making use of the Giraud-Rezk-Lurie theorem), the second elementary. (Following this discussion.)

Proof

Observe that n\sharp_n preserves products, since \sharp does (being a right adjoint) and by this proposition. Now use that the homotopy quotient V/GV/G is the realization of the simplicial object (V/G) =G × ×V(V/G)_\bullet = G^{\times_{\bullet}} \times V. So applying n\sharp_n to this yields a simplicial object (( nV)/( nG)) =( nG) × ×( nV)((\sharp_n V)/(\sharp_n G))_\bullet = (\sharp_n G)^{\times_{\bullet}} \times (\sharp_n V) which exhibits the desired action.

Proof

Generally, let A:BTypeA:B\to Type be any dependent type family (speaking homotopy type theory). We claim that there is an induced family A n: n+1BTypeA^{\sharp_n} : \sharp_{n+1} B \to Type such that A n(η n+1(b))= n(A(b))A^{\sharp_n}(\eta_{n+1}(b)) = \sharp_n (A(b)) for any b:Bb:B, where η n+1:B n+1B\eta_{n+1} : B \to \sharp_{n+1} B is the inclusion. Applying this when ABA \to B is V/GBGV/G \to \mathbf{B}G and when bb is (necessarily) the basepoint of BG\mathbf{B}G gives the desired action on the desired type.

First of all, we have the composite BATypeType B \xrightarrow{A} Type \xrightarrow{\sharp} Type_{\sharp}, where Type = X:Typeis(X)Type_{\sharp} = \sum_{X:Type} is\sharp(X). Since Type Type_{\sharp} is itself \sharp (since \sharp is lex), this factors through B\sharp B, giving a type family A :BType A^\sharp : \sharp B \to Type_{\sharp} such that A (η(b))=(A(b))A^{\sharp}(\eta(b)) = \sharp (A(b)) for any b:Bb:B, where η:BB\eta:B\to \sharp B is the unit of \sharp.

Now fix y:By:\sharp B and x:A (y)x:A^\sharp(y). For any b:Bb:B and p:η(b)=yp:\eta(b)=y, we can define the type (a:A(b))p *(η(a))=x n{\big\Vert \sum_{(a:A(b))} p_\ast (\eta(a)) = x\big\Vert}_n. This is an nn-type, and since the type of truncated types n-Typen\text{-}Type is an (n+1)(n+1)-type, as a function of (b,p): b:Bη(b)=y(b,p) : \sum_{b:B} \eta(b)=y, this construction factors through b:Bη(b)=y n+1\big\Vert \sum_{b:B} \eta(b)=y\big\Vert_{n+1}. Thus, for y:By:\sharp B and x:A (y)x:A^\sharp(y) and ξ: (b:B)η(b)=y n+1\xi : {\big\Vert \sum_{(b:B)} \eta(b)=y\big\Vert}_{n+1} we have a type P(y,x,ξ)P(y,x,\xi), such that

P(y,x,|(b,p)| n+1)= (a:A(b))p *(η(a))=x n.P\big(y,x,{|(b,p)|}_{n+1}\big) = {\left\Vert \sum_{(a:A(b))} p_\ast (\eta(a)) = x\right\Vert}_n.

Now by definition, n+1B (y:B) (b:B)η(b)=y n+1\sharp_{n+1} B \coloneqq \sum_{(y:\sharp B)} {\big\Vert \sum_{(b:B)} \eta(b)=y\big\Vert}_{n+1}. Thus, we can define A n: n+1BTypeA^{\sharp_n} : \sharp_{n+1} B \to Type by A n(y,ξ)= x:A (y)P(y,x,ξ)A^{\sharp_n}(y,\xi) = \sum_{x:A^\sharp(y)} P(y,x,\xi). And since η n+1(b)=(η(b),|(b,1)| n+1)\eta_{n+1}(b) = (\eta(b),{|(b,1)|}_{n+1}), we have A n(η n+1(b))= x:(A(b)) (a:A(b))η(a))=x nA^{\sharp_n}(\eta_{n+1}(b)) = \sum_{x:\sharp(A(b))} {\big\Vert \sum_{(a:A(b))} \eta(a)) = x\big\Vert}_n, which is n(A(b))\sharp_{n}(A(b)) by definition.

Infinitesimally: actions of L L_\infty-algebroids

See Lie infinity-algebroid representation.

Properties

Model category presentation

In the context of geometrically discrete ∞-groupoids a model category structure presenting the (∞,1)-category of \infty-actions is the Borel model structure (DDK 80).

representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory:

homotopy type theoryrepresentation theory
pointed connected context BG\mathbf{B}G∞-group GG
dependent type∞-action/∞-representation
dependent sum along BG*\mathbf{B}G \to \astcoinvariants/homotopy quotient
context extension along BG*\mathbf{B}G \to \asttrivial representation
dependent product along BG*\mathbf{B}G \to \asthomotopy invariants/∞-group cohomology
dependent product of internal hom along BG*\mathbf{B}G \to \astequivariant cohomology
dependent sum along BGBH\mathbf{B}G \to \mathbf{B}Hinduced representation
context extension along BGBH\mathbf{B}G \to \mathbf{B}H
dependent product along BGBH\mathbf{B}G \to \mathbf{B}Hcoinduced representation
spectrum object in context BG\mathbf{B}Gspectrum with G-action (naive G-spectrum)

References

General

Actions of A-∞ algebras in some symmetric monoidal (∞,1)-category are discussed in section 4.2 of

Aspects of actions of ∞-groups in an ∞-topos in the contect of associated ∞-bundles are discussed in section I 4.1 of

For discrete geometry

For H=Grpd\mathbf{H}= \infty Grpd the statement that homotopy types over BGB G are equivalently GG-infinity-actions is (via the Borel model structure) due to

  • E. Dror, William Dwyer, Daniel Kan, Equivariant maps which are self homotopy equivalences, Proc. Amer. Math. Soc. 80 (1980), no. 4, 670–672 (JSTOR)

This is mentioned for instance as exercise 4.2 in

  • William Dwyer, Homotopy theory of classifying spaces, Lecture notes Copenhagen (June, 2008) pdf

Closely related discussion of homotopy fiber sequences and homotopy action but in terms of Segal spaces is in section 5 of

There, conditions are given for a morphism A B A_\bullet \to B_\bullet to a reduced Segal space to have a fixed homotopy fiber, and hence encode an action of the loop group of BB on that fiber.

For actions of topological groups

That GG-actions for GG a topological group in the sense of G-spaces in equivariant homotopy theory (and hence with GG not regarded as the geometrically discrete ∞-group of its underying homotopy type ) are equivalently objects in the slice (∞,1)-topos over BG\mathbf{B}G is Elmendorf's theorem together with the fact, highlighted in this context in

that

GSpacePSh (Orb G)PSh (Orb /BG)PSh (Orb) /BG G Space \simeq PSh_\infty(Orb_G) \simeq PSh_\infty(Orb_{/\mathbf{B}G}) \simeq PSh_\infty(Orb)_{/\mathbf{B}G}

is therefore the slice of the \infty-topos over the global orbit category by BG\mathbf{B}G.

Rezk-global equivariant homotopy theory:

cohesive (∞,1)-toposits (∞,1)-sitebase (∞,1)-toposits (∞,1)-site
global equivariant homotopy theory PSh (Glo)PSh_\infty(Glo)global equivariant indexing category GloGlo∞Grpd PSh (*) \simeq PSh_\infty(\ast)point
sliced over terminal orbispace: PSh (Glo) /𝒩PSh_\infty(Glo)_{/\mathcal{N}}Glo /𝒩Glo_{/\mathcal{N}}orbispaces PSh (Orb)PSh_\infty(Orb)global orbit category
sliced over BG\mathbf{B}G: PSh (Glo) /BGPSh_\infty(Glo)_{/\mathbf{B}G}Glo /BGGlo_{/\mathbf{B}G}GG-equivariant homotopy theory of G-spaces L weGTopPSh (Orb G)L_{we} G Top \simeq PSh_\infty(Orb_G)GG-orbit category Orb /BG=Orb GOrb_{/\mathbf{B}G} = Orb_G

See at equivariant homotopy theory for more references along these lines.

Revised on June 5, 2015 06:51:34 by Urs Schreiber (50.207.161.2)