nLab
infinity-action

Idea

The notion of $\mathrm{\infty }$-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 $A$ may act on another object $N$ by a morphism $A\otimes N\to N$ which satisfies an action property up to coherent higher homotopy.

If the $\mathrm{\infty }$-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 $H$ be an (∞,1)-topos.

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

$$(\cdots \overrightarrow{\overrightarrow{\overrightarrow{\to }}}G\times G\overrightarrow{\overrightarrow{\to }}G\overrightarrow{\to }*)$$\left( \cdots \stackrel{\to}{\stackrel{\to}{\stackrel{\to}{\to}}} G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} * \right)

satisfying the groupoidal Segal conditions.

hence an ∞-group.

Definition

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

$$(\cdots \overrightarrow{\overrightarrow{\overrightarrow{\to }}}V\times G\times G\overrightarrow{\overrightarrow{\to }}V\times G\stackrel{\stackrel{\rho }{\to }}{\underset{p_1}{\to }}V)$$\left( \cdots \stackrel{\to}{\stackrel{\to}{\stackrel{\to}{\to}}} V \times G \times G \stackrel{\to}{\stackrel{\to}{\to}} V \times G \stackrel{\overset{\rho}{\to}}{\underset{p_1}{\to}} V \right)

such that the projection maps

$$\begin{array}{ccccccc}\cdots & \overrightarrow{\overrightarrow{\overrightarrow{\to }}}& V\times G\times G& \overrightarrow{\overrightarrow{\to }}& V\times G& \stackrel{\stackrel{\rho }{\to }}{\underset{p_1}{\to }}& V\\ & & \downarrow & & \downarrow & & \downarrow \\ \cdots & \overrightarrow{\overrightarrow{\overrightarrow{\to }}}& G\times G& \overrightarrow{\overrightarrow{\to }}& G& \overrightarrow{\underset{}{\to }}& *\end{array}$$\array{ \cdots &\stackrel{\to}{\stackrel{\to}{\stackrel{\to}{\to}}}& V \times G \times G &\stackrel{\to}{\stackrel{\to}{\to}}& V \times G &\stackrel{\overset{\rho}{\to}}{\underset{p_1}{\to}}& V \\ && \downarrow && \downarrow && \downarrow \\ \cdots &\stackrel{\to}{\stackrel{\to}{\stackrel{\to}{\to}}}& G \times G &\stackrel{\to}{\stackrel{\to}{\to}}& G &\stackrel{\overset{}{\to}}{\underset{}{\to}}& * }

constitute a morphism of groupoid objects $V⫽G\to *⫽G$.

The (∞,1)-category of such actions is the slice of groupoid objects over $*⫽G$ 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

$$B:\mathrm{Grp}(H)\to H_{\ge 1}^{*/}.$$\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 $H$.

Proposition

Every $\mathrm{\infty }$-action $\rho :V\times G\to V$ has a classifying morphism $c_{\rho }:V//G\to BG$ in that there is a fiber sequence

$$\begin{array}{c}V\\ \downarrow \\ V//G& \stackrel{\overline{\rho }}{\to }& BG\end{array}$$\array{ V \\ \downarrow \\ V//G &\stackrel{\overline{\rho}}{\to}& \mathbf{B}G }

such that $\rho$ is the $G$-action on $V$ regarded as the corresponding $G$-principal ∞-bundle modulated by $\overline{\rho }$.

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

Definition

For $V\in H$ an object, a $G$-$\mathrm{\infty }$-action $\rho$ on $V$ is a fiber sequence in $H$ of the form

$$\begin{array}{ccc}V& \to & V⫽G\\ & & {\downarrow }^{\overline{\rho }}\\ & & BG\end{array}.$$\array{ V &\to& V \sslash G \\ && \downarrow^{\mathrlap{\overline{\rho}}} \\ && \mathbf{B}G } \,.

The (∞,1)-category of $G$-actions in $H$ is the slice (∞,1)-topos of $H$ over $BG$:

$${\mathrm{Act}}_H(G)≔H_{/BG}.$$Act_{\mathbf{H}}(G) \coloneqq \mathbf{H}_{/\mathbf{B}G} \,.

Remark

A $\rho \in {\mathrm{Act}}_H(G)$ corresponds to a morphism denoted $\overline{\rho }:V⫽G\to BG$ in $H$ hence to an object $\overline{\rho }\in H_{/BG}$.

A morphism $\varphi :{\rho }_1\to {\rho }_2$ in ${\mathrm{Act}}_H(G)$ corresponds to a diagram

$$\begin{array}{ccccc}{V}_1⫽G& & \stackrel{}{\to }& & V_2⫽G\\ & {}_{\overline{{\rho }_1}}\searrow & & {\swarrow }_{\overline{{\rho }_2}}\\ & & BG\end{array}$$\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 $H$.

Remark

The bundle $\overline{\rho }$ in def. 2 is the universal $\rho$-associated $V$-fiber ∞-bundle.

Remark

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

$$*:BG⊢V(*):\mathrm{Type}.$$* : \mathbf{B}G \vdash V(*) : Type \,.

Notions in higher representation theory

We discuss some basic representation theoretic notions of $\mathrm{\infty }$-actions.

In summary, for $c:BG⊢V(c):\mathrm{Type}$ an action of $G$ on $V$, we have

• the dependent sum

  $$⊢\sum _{c:BG}V(c):\mathrm{Type}$$\vdash \sum_{\mathbf{c} : \mathbf{B}G} V(\mathbf{c}) : Type

  is the quotient $V⫽G$ of $V$ by $G$;

• the dependent product

  $$⊢\prod _{c:BG}V(c):\mathrm{Type}$$\vdash \prod_{\mathbf{c} : \mathbf{B}G} V(\mathbf{c}) : Type

  is the collection of invariants of the actions.

And for $V_1,V_2$ two actions we have

• the dependent product over the dependent function type

  $$⊢:\prod _{c:BG}(V_1(c)\to V_2(c)):\mathrm{Type}$$\vdash : \prod_{\mathbf{c} : \mathbf{B}G} (V_1(\mathbf{c}) \to V_2(\mathbf{c})) : Type

  is the collection of $G$-homomorphisms ($G$-equivariant maps);

• the dependent sum over the dependent function type

  $$⊢:\sum _{c:BG}(V_1(c)\to V_2(c)):\mathrm{Type}$$\vdash : \sum_{\mathbf{c} : \mathbf{B}G} (V_1(\mathbf{c}) \to V_2(\mathbf{c})) : Type

  is the quotient of all functions $V_1\to V_2$ by the conjugation action of $G$.

Invariants

Definition

The invariants of a $G$-$\mathrm{\infty }$-action are the sections of the morphism $V⫽G\to BG$,

$$\mathrm{Invariants}(V)=\prod _{BG\to *}(V⫽G\to BG),$$Invariants(V) = \prod_{\mathbf{B}G \to *} (V \sslash G \to \mathbf{B}G) \,,

where ${\prod }_{BG\to *}:H_{/BG}\to H$ is the direct image of the base change geometric morphism.

In homotopy type theory syntax: for

$$c:BG⊢V(c):\mathrm{Type}$$\mathbf{c} : \mathbf{B}G \vdash V(\mathbf{c}) : Type

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

$$⊢\prod _{c:BG}V(c):\mathrm{Type}.$$\vdash \prod_{\mathbf{c} : \mathbf{B}G} V(\mathbf{c}) : Type \,.

Quotients

From def. 2 we read off:

Definition

The quotient of a $G$-action

$$c:BG⊢V(c):\mathrm{Type}$$\mathbf{c} : \mathbf{B}G \vdash V(\mathbf{c}) : Type

is the dependent sum

$$⊢\sum _{c:BG}V(c):\mathrm{Type}.$$\vdash \sum_{\mathbf{c} : \mathbf{B}G} V(\mathbf{c}) : Type \,.

Conjugation actions

Remark

By def. 2, and basic facts disussed at slice (∞,1)-topos, the (∞,1)-category ${\mathrm{Act}}_H(G)$ 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 $\mathrm{\infty }$-actions given this way. The following statements are essentially immediate consequences of basic homotopy type theory.

Proposition

For $(V_1,{\rho }_1),(V_2,{\rho }_2)\in \mathrm{Act}(G)$ their cartesian product is a $G$-action on the product of $V_1$ with $V_2$ in $H$.

Proof

Let

$$\begin{array}{ccc}{V}_i& \to & V⫽G\\ & & {\downarrow }^{{\overline{\rho }}_i}\\ & & BG\end{array}$$\array{ V_i &\to& V\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 $(V_1,{\rho }_1)\times (V_2,{\rho }_2)\in \mathrm{Act}(G)$ is given in $H$ by the (∞,1)-pullback

$$\sum _{BG}{\overline{\rho }}_1\times {\overline{\rho }}_2\simeq V_1⫽G{\times }_{BG}{V}_2⫽G$$\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 $H$, with the product action being exhibited by the principal ∞-bundle

$$\begin{array}{ccc}{V}_1\times V_2& \to & V_1⫽G{\times }_{BG}{V}_2⫽G\\ & & {\downarrow }^{\overline{{\rho }_1\times {\rho }_2}}\\ & & BG\end{array}.$$\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 $V_1\times V_2$ by using that (∞,1)-limits commute over each other.

Proposition

For ${\rho }_1,{\rho }_2\in \mathrm{Act}(G)$ their internal hom $[{\rho }_1,{\rho }_2]\in {\mathrm{Act}}_H(G)$ is a $G$-action on the internal hom $[V_1,V_2]\in H$.

Proof

Taking fibers

$${\mathrm{pt}}_{BG}^{*}:H_{/BG}\to H$$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:

$$\begin{array}{rl}{\mathrm{pt}}_{BG}^{*}[{\overline{\rho }}_1,{\overline{\rho }}_2]& \simeq [{\mathrm{pt}}_{BG}^{*}{\overline{\rho }}_1,{\mathrm{pt}}_{BG}^{*}{\overline{\rho }}_2]\\ & \simeq [V_1,V_2]\end{array}.$$\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

$$\begin{array}{ccc}[V_1,V_2]& \to & V_1⫽G{\times }_{BG}{V}_2⫽G\\ & & {\downarrow }^{\overline{[{\rho }_1,{\rho }_2]}}\\ & & BG\end{array}$$\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 $G$ on $[V_1,V_2]$ by pre- and post-composition of functions $V_1\to V_2$ with the $G$-action on $V_1$ and on $V_2$, 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 ${\overline{\rho }}_i:V_i⫽G\to BG$ two $G$-actions, the object of homomorphisms is

$$\prod _{BG\to *}[{\overline{\rho }}_1,{\overline{\rho }}_2]\in H.$$\prod_{\mathbf{B}G \to *}[\bar \rho_1, \bar \rho_2] \in \mathbf{H} \,.

In the syntax of homotopy type theory

$$⊢\prod _{c:BG}{V}_1(c)\to V_2(c):\mathrm{Type}.$$\vdash \prod_{\mathbf{c} : \mathbf{B}G} V_1(\mathbf{c}) \to V_2(\mathbf{c}) : Type \,.

Stabilizer subgroups

See at stabilizer subgroup.

Examples

Of $\mathrm{\infty }$-group actions in an $\mathrm{\infty }$-topos

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

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

Trivial action

Consider the étale geometric morphism

$${\mathrm{Act}}_H(G)≔H_{/BG}\stackrel{\stackrel{p^{*}≔(-)\times BG}{\leftarrow }}{\underset{}{\to }}H.$$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 $V\in H$ any object, the trivial action of $G$ on $V$ is $p^{*}V\in {\mathrm{Act}}_H(G)$, exhibited by the split fiber sequence

$$\begin{array}{ccc}V& \to & V\times BG\\ & & \downarrow \\ & & BG\end{array}.$$\array{ V &\to& V \times \mathbf{B}G \\ && \downarrow \\ && \mathbf{B}G } \,.

Fundamental action

The right $\mathrm{\infty }$-action of $G$ on itself is given by the fiber sequence

$$\begin{array}{c}G\\ \downarrow \\ *& \to & BG\end{array}$$\array{ G \\ \downarrow \\ * &\to& \mathbf{B}G }

which exhibits $BG$ as the delooping of $G$.

$$G//G\simeq *.$$G//G \simeq * \,.

Adjoint action

The fiber sequence

$$\begin{array}{c}G\\ \downarrow \\ ℒBG& \stackrel{{\mathrm{ev}}_{*}}{\to }& BG\end{array}$$\array{ G \\ \downarrow \\ \mathcal{L} \mathbf{B}G &\stackrel{ev_*}{\to}& \mathbf{B}G }

given by the free loop space object $ℒBG$ exhibits the higher adjoint action of $G$ on itself:

$$G/{/}_{\mathrm{Ad}}G\simeq ℒBG.$$G//_{Ad}G \simeq \mathcal{L}\mathbf{B}G \,.

Automorphism action

Definition

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

$$\begin{array}{c}V\\ \downarrow \\ V//\mathrm{Aut}(V)& \to & B\mathrm{Aut}(V)\end{array}$$\array{ V \\ \downarrow \\ V//\mathbf{Aut}(V) &\to& \mathbf{B} \mathbf{Aut}(V) }

Conjugation actions

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

This example spells out everything completely in components:

Example

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

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

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

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

Hence a morphism of $G$-actions

$$\varphi :V\to [\Sigma ,X]$$\phi : V \to [\Sigma,X]

is a function $\varphi$ of the underlying sets such that for all $V\in V$, $g\in G$ and all $\sigma \in \Sigma$ we have

(1)$$\varphi (g(v))(\sigma )=g(\varphi (v)(g^{-1}(\sigma )).$$\phi(g(v))(\sigma) = g(\phi(v)(g^{-1}(\sigma)) \,.

On the other hand, a morphism of actions

$$\psi :V\times \Sigma \to X$$\psi : V \times \Sigma \to X

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

$$\psi (g(v),g(\sigma ))=g(\psi (v,\sigma ))$$\psi(g(v), g(\sigma)) = g(\psi(v,\sigma))

which is equivalent to

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

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

$$\psi (v,\sigma )≔\varphi (v)(\sigma )$$\psi(v,\sigma) \coloneqq \phi(v)(\sigma)

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

$${\mathrm{Act}}_H(G)(V,[\Sigma ,X])\simeq {\mathrm{Act}}_H(G)(V\times \Sigma ,X]),$$Act_{\mathbf{H}}(G)(V, [\Sigma,X]) \simeq Act_{\mathbf{H}}(G)(V \times \Sigma, X]) \,,

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

Remark

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

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

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

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

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

General covariance

Let $X\in H$ be a moduli infinity-stack for field in a gauge theory or sigma-model. Let $\Sigma \in H$ be the corresponding spacetime or worldvolume, respectively.

We have the automorphism action, def. 7

$$\begin{array}{ccc}\Sigma & \to & \Sigma ⫽\mathrm{Aut}(\Sigma )\\ & & \downarrow \\ & & B\mathrm{Aut}(\Sigma )\end{array}.$$\array{ \Sigma &\to& \Sigma \sslash \mathbf{Aut}(\Sigma) \\ && \downarrow \\ && \mathbf{B} \mathbf{Aut}(\Sigma) } \,.

The slice $H_{/\mathrm{Aut}(\Sigma )}={\mathrm{Act}}_H(\mathrm{Aut}(\Sigma ))$ is the context of types which are generally covariant over $\Sigma$.

On $X$ consider the trivial $\mathrm{Aut}(\Sigma )$-action, def. 6. Then the internal-hom action of prop. 3

$$[\Sigma ,X]⫽\mathrm{Aut}(\Sigma )\simeq [\Sigma ⫽\mathrm{Aut}(\Sigma ),X\times B\mathrm{Aut}(\Sigma ){]}_{B\mathrm{Aut}(\Sigma )}$$[\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 $\Sigma$ modulo automorphisms (diffeomorphisms, in smooth cohesion) of $\Sigma$. This is the configuration space of “generally covariant” field theory on $\Sigma$.

Semidirect product groups

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

$$\begin{array}{ccc}BA& \to & B(G⋉A)\\ & & \downarrow \\ & & BG\end{array}.$$\array{ \mathbf{B}A &\to& \mathbf{B} (G \ltimes A) \\ && \downarrow \\ && \mathbf{B}G } \,.

The corresponding action groupoid $(BA)⫽G\simeq B(G⋉A)$ is the delooping of the corresponding semidirect product group.

$G$-Modules

Definition

For $G\in \mathrm{Grp}(H)$ the $\mathrm{\infty }$-category of $G$-modules is

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

the stabilization of the $\mathrm{\infty }$-category of $G$-actions.

Example

For $G$ and $A$ 0-truncated groups, $A$ an abelian group with $G$-module structure, the semidirect product group $G⋉A$ from above exhibits $A$ as a $G$-module in the sense of def. 8.

Related concepts

• action, $\mathrm{\infty }$-action

• module, ∞-module

• representation, ∞-representation

• associated bundle, associated ∞-bundle

• induced representation

• group cohomology

References

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

• Jacob Lurie, Higher Algebra

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

• Principal ∞-bundles -- theory, presentations and applications