# nLab infinity-action

### Context

#### Higher algebra

higher algebra

universal algebra

# 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 $A$ 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 $\infty$-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 $\mathbf{H}$ be an (∞,1)-topos.

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

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

satisfying the groupoidal Segal conditions.

hence an ∞-group.

###### Definition

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

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

such that the projection maps

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

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

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

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

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

###### Proposition

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

$\array{ V \\ \downarrow \\ V \sslash G &\stackrel{\overline{\rho}}{\to}& \mathbf{B}G }$

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

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

###### Definition

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

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

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

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

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

A morphism $\phi : \rho_1 \to \rho_2$ in $Act_{\mathbf{H}}(G)$ corresponds to a diagram

$\array{ V_1 \sslash G &&\stackrel{}{\to}&& V_2 \sslash G \\ & {}_{\mathllap{\overline{\rho_1}}}\searrow && \swarrow_{\mathrlap{\overline{\rho_2}}} \\ && \mathbf{B}G }$

in $\mathbf{H}$.

###### Remark

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

###### Remark

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

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

## Notions in higher representation theory

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

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

• $\vdash \sum_{\mathbf{c} : \mathbf{B}G} V(\mathbf{c}) : Type$

is the quotient $V\sslash G$ of $V$ by $G$;

• $\vdash \prod_{\mathbf{c} : \mathbf{B}G} V(\mathbf{c}) : Type$

is the collection of invariants (homotopy fixed points) of the action.

And for $V_1, V_2$ two actions we have

• the dependent product over the dependent function 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

$\vdash \sum_{\mathbf{c} : \mathbf{B}G} (V_1(\mathbf{c}) \to V_2(\mathbf{c})) : Type$

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

### Invariants

###### Definition

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

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

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

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

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

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

### Quotients

From def. 2 we read off:

###### Definition

The quotient of a $G$-action

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

is the dependent sum

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

### Conjugation actions

###### Remark

By def. 2, and basic facts disussed at slice (∞,1)-topos, the (∞,1)-category $Act_{\mathbf{H}}(G)$ 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 $(V_1, \rho_1), (V_2, \rho_2) \in Act(G)$ their cartesian product is a $G$-action on the product of $V_1$ with $V_2$ in $\mathbf{H}$.

###### Proof

Let

$\array{ V_i &\to& V_i \sslash G \\ && \downarrow^{\bar \rho_i} \\ && \mathbf{B}G }$

be the principal ∞-bundles exhibiting the two actions.

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

$\sum_{\mathbf{B}G} \bar \rho_1 \times \bar \rho_2 \simeq V_1\sslash G \times_{\mathbf{B}G} V_2 \sslash G$

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

$\array{ V_1 \times V_2 &\to& V_1\sslash G \times_{\mathbf{B}G} V_2 \sslash G \\ && \downarrow^{\mathrlap{\overline{ \rho_1 \times \rho_2 }}} \\ && \mathbf{B}G } \,.$

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

###### Proof

Taking fibers

$pt_{\mathbf{B}G}^* : \mathbf{H}_{/\mathbf{B}G} \to \mathbf{H}$

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{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

$\array{ [V_1,V_2] &\to& V_1 \sslash G \times_{\mathbf{B}G} V_2 \sslash G \\ && \downarrow^{\mathrlap{\overline{[\rho_1,\rho_2]}}} \\ && \mathbf{B}G }$

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.

### 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 $\bar \rho_i : V_i \sslash G \to \mathbf{B}G$ two $G$-actions, the object of homomorphisms is

$\prod_{\mathbf{B}G \to *}[\bar \rho_1, \bar \rho_2] \in \mathbf{H} \,.$

In the syntax of homotopy type theory

$\vdash \prod_{\mathbf{c} : \mathbf{B}G} V_1(\mathbf{c}) \to V_2(\mathbf{c}) : Type \,.$

### Stabilizer subgroups

See at stabilizer subgroup.

## Examples

### Of $\infty$-group actions in an $\infty$-topos

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

The following lists some fundamental classes of examples of $\infty$-actions of $G$, 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

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

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

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

#### Fundamental action

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

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

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

$G \sslash G \simeq * \,.$

The fiber sequence

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

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

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

#### Automorphism action

###### Definition

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

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

#### Conjugation actions

We discuss the simple case of the cartesian closed category of $G$-sets (G-permutation representations) for $G$ 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 $\mathbf{H} =$ ∞Grpd, let $G \in Grp(\infty Grpd)$ 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

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

(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

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

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

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

On the other hand, a morphism of actions

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

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

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

which is equivalent to

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

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

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

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

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

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

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

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

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

#### General covariance

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

We have the automorphism action, def. 7

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

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

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

$[\Sigma, X]\sslash \mathbf{Aut}(\Sigma) \simeq [\Sigma \sslash \mathbf{Aut}(\Sigma), X \times \mathbf{B}\mathbf{Aut}(\Sigma)]_{\mathbf{B}\mathbf{Aut}(\Sigma)}$

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 $G, A \in Grp(\mathbf{H})$ 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 $\mathbf{B}A$, hence a fiber sequence

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

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

#### $G$-Modules

###### Definition

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

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

the stabilization of the $\infty$-category of $G$-actions.

###### Example

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

homotopy type theoryrepresentation theory
pointed connected context $\mathbf{B}G$∞-group $G$
dependent type∞-action/∞-representation
dependent sum along $\mathbf{B}G \to \ast$coinvariants/homotopy quotient
context extension along $\mathbf{B}G \to \ast$trivial representation
dependent product along $\mathbf{B}G \to \ast$homotopy invariants/∞-group cohomology
dependent sum along $\mathbf{B}G \to \mathbf{B}H$induced representation
context extension along $\mathbf{B}G \to \mathbf{B}H$
dependent product along $\mathbf{B}G \to \mathbf{B}H$coinduced representation

## References

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

Revised on November 16, 2013 07:49:39 by Urs Schreiber (89.204.155.122)