nLab infinity-action

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, following NSS. (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

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

satisfying the groupoidal Segal conditions.

hence an ∞-group.

Definition

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

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

such that the projection maps

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

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

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

At least in the special case that $\mathbf{H} \,=\, Grpd_\infty$, this may also be understood as an instance of the “fundamental theorem of $\infty$-topos theory”, see there. For a model category-presentation see also Borel model structure – Relation to the slice over the simplicial classifying space.

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. is the universal $\rho$-associated $V$-fiber ∞-bundle.

Remark

In the form of def. $\infty$-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

• the dependent sum

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

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

• the dependent product

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

  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 $\rho$ 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.

In homotopy type theory syntax for

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

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

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

Remark

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

$$\rho \colon \mathbf{B}G \to Type \,.$$

See at internal limit – Examples – Homotopy Invariants.

Coinvariants / Quotients

From def. we read off:

Definition

The quotient of a $G$-action

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

is the dependent sum

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

Remark

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

$$\rho \colon \mathbf{B}G \to Type \,.$$

See at internal limit – Examples – Homotopy Coinvariants.

Conjugation actions

Remark

By def. , 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.

See also at Conjugation actions below.

Internal object of homomorphisms

Remark

The invariant, def. of the conjugation action, prop. 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 groups

See at stabilizer group.

Linearization

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

Let $0 \colon \ast \to V$ be a pointed object.

Let $G$ be an $\infty$-group acting on $V$

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

such that this action preserves the point of $V$, 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

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

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

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

(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

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

exhibiting that the $G$-action on $V$ restricts to the trivial action on the point $0$ of $V$.

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

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

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

Here the cartesian subdiagram

$$\array{ \mathbb{D}^V_0 &\longrightarrow& \mathbb{D}^V_0/G \\ \downarrow && \downarrow \\ \ast &\longrightarrow& \mathbf{B}G }$$

hence exhibits a $G$-action on $\mathbb{D}^V_0$.

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

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

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

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

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

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

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

Definition

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

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

then the corresponding action groupoid is

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

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

Schematically:

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

Example

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

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

This makes it clear that:

Proposition

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

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

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

This morphism is an isofibration.

Proposition

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

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

Proof

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

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

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

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

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

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

Proposition

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

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

Proof

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

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

Definition

Given a homotopy fiber sequence of groupoids of the form

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

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

Consider the homotopy fiber product

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

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

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

there is a canonical equivalence of groupoids

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

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

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

Remark

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

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

and hence, via the construction in def. , by

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

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

Proposition

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

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

commutes.

Proposition

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

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

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

This equivalence takes an action to its action groupoid.

Proof

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

$\infty$-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 $G$.

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

Adjoint action

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:

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

For more on this see at free loop space of a classifying space.

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. .

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

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

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

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.

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

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. . Then the internal-hom action of prop.

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

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. .

Actions in a slice

Consider an object $B \in \mathbf{H}$ and an object

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

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

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

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

$$\epsilon \colon B^\ast \mathbf{B}\mathbf{Aut}_{\mathbf{H}}(L) \longrightarrow \mathbf{B} \mathbf{Aut}(L) \,.$$

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

$$\array{ L &\longrightarrow& L//(B^\ast\mathbf{Aut}_{\mathbf{H}}(L)) &\longrightarrow& L//\mathbf{Aut}(L) \\ \downarrow && \downarrow && \downarrow \\ \ast &\longrightarrow& \mathbf{B}B^\ast \mathbf{Aut}_{\mathbf{H}}(L) &\stackrel{\epsilon}{\longrightarrow}& \mathbf{B} \mathbf{Aut}(L) }$$

Proposition

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

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

and

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

Proof

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

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

yields

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

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

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

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

Remark

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

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

act on $X$ by the evident restriction to the horizontal equivalences,

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

and that forming the homotopy quotient of this action on $L$ makes $L$ descent to the homotopy quotient of $X$ by this action to yield

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

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

Proposition

Given an ∞-group $G$ in $\mathbf{H}$ and a $G$-action, def. , on some $X$, then $\sharp_n G$ is itself canonically an $\infty$-group equipped with a canonically induced action on $\sharp_n X$ such that the projection $X \to \sharp_n X$ carries the structure of a homomorphism of $G$-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 $\sharp_n$ preserves products, since $\sharp$ does (being a right adjoint) and by this proposition. Now use that the homotopy quotient $V/G$ is the realization of the simplicial object $(V/G)_\bullet = G^{\times_{\bullet}} \times V$. So applying $\sharp_n$ to this yields a simplicial object $((\sharp_n V)/(\sharp_n G))_\bullet = (\sharp_n G)^{\times_{\bullet}} \times (\sharp_n V)$ which exhibits the desired action.

Proof

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

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

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

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

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

Infinitesimally: actions of $L_\infty$-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).

Related concepts

• action, $\infty$-action

  • faithful ∞-action

  • stabilizer ∞-group

• simplicial group action, topological G-space

  model structure on simplicial group actions

• module, ∞-module

• representation, ∞-representation

• associated bundle, associated ∞-bundle

• induced representation

• equivariant homotopy theory

• equivariant cohomology

  • group cohomology

representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory (FSS 12 I, exmp. 4.4):

homotopy type theory	representation theory
pointed connected context $\mathbf{B}G$	∞-group $G$
dependent type on $\mathbf{B}G$	$G$-∞-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 product of internal hom along $\mathbf{B}G \to \ast$	equivariant cohomology
dependent sum along $\mathbf{B}G \to \mathbf{B}H$	induced representation
context extension along $\mathbf{B}G \to \mathbf{B}H$	restricted representation
dependent product along $\mathbf{B}G \to \mathbf{B}H$	coinduced representation
spectrum object in context $\mathbf{B}G$	spectrum 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

• Jacob Lurie, Higher Algebra

The discussion of $\infty$-actions as presented above follows:

• Thomas Nikolaus, Urs Schreiber, Danny Stevenson, I §4.1 of: Principal ∞-bundles – theory, presentations and applications, Journal of Homotopy and Related Structures (2014) [arXiv:1207.0248]

• Hisham Sati, Urs Schreiber: §2.2 of: Proper Orbifold Cohomology [arXiv:2008.01101]

• Hisham Sati, Urs Schreiber: §3.2.3 in: Equivariant principal $\infty$-bundles [arXiv:2112.13654]

Discussion in homotopy type theory:

• Ulrik Buchholtz, Floris van Doorn, Egbert Rijke, Higher Groups in Homotopy Type Theory, LICS ‘18: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (arXiv:1802.04315, doi:10.1145/3209108.3209150)

• Ulrik Buchholtz, Notes on higher groups and projective spaces, 2016 (pdf)

On extensions of actions on a space to actions on geometric structures defined over it:

• Severin Bunk, Carlos S. Shahbazi. Higher Geometric Structures on Manifolds and the Gauge Theory of Deligne Cohomology (2023). (arXiv:2304.06633).

For discrete geometry

For $\mathbf{H}= \infty Grpd$ the statement that homotopy types over $B G$ are equivalently $G$-infinity-actions is (via the Borel model structure) is 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:2043448)

This is mentioned for instance as exercise 4.2 in

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

An alternative proof in terms of relative categories is in

• Amit Sharma, On the homotopy theory of $G$-spaces, International Journal of Mathematics and Statistics Invention, Volume7 Issue 2, 2019 (arXiv:1512.03698, published pdf)

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

• Matan Prezma, Homotopy normal maps, Algebr. Geom. Topol. Volume 12, Number 2 (2012), 1211-1238. (arXiv:1011.4708, euclid:agt/1513715387)

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

For actions of topological groups

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

• Charles Rezk, Global Homotopy Theory and Cohesion, 2014

that

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

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

Rezk-global equivariant homotopy theory:

cohesive (∞,1)-topos	its (∞,1)-site	base (∞,1)-topos	its (∞,1)-site
global equivariant homotopy theory $PSh_\infty(Glo)$	global equivariant indexing category $Glo$	∞Grpd $\simeq PSh_\infty(\ast)$	point
… sliced over terminal orbispace: $PSh_\infty(Glo)_{/\mathcal{N}}$	$Glo_{/\mathcal{N}}$	orbispaces $PSh_\infty(Orb)$	global orbit category
… sliced over $\mathbf{B}G$: $PSh_\infty(Glo)_{/\mathbf{B}G}$	$Glo_{/\mathbf{B}G}$	$G$-equivariant homotopy theory of G-spaces $L_{we} G Top \simeq PSh_\infty(Orb_G)$	$G$-orbit category $Orb_{/\mathbf{B}G} = Orb_G$

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