symmetric monoidal (∞,1)-category of spectra
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.
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
satisfying the groupoidal Segal conditions.
hence an ∞-group.
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
such that the projection maps
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
form group objects in an (∞,1)-category to the (∞,1)-category of connected pointed objects in $\mathbf{H}$.
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
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.
For $V \in \mathbf{H}$ an object, a $G$-$\infty$-action $\rho$ on $V$ is a fiber sequence in $\mathbf{H}$ of the form
The (∞,1)-category of $G$-actions in $\mathbf{H}$ is the slice (∞,1)-topos of $\mathbf{H}$ over $\mathbf{B}G$:
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
in $\mathbf{H}$.
The bundle $\overline{\rho}$ in def. 2 is the universal $\rho$-associated $V$-fiber ∞-bundle.
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 $V$:
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
is the quotient $V\sslash G$ of $V$ by $G$;
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
is the collection of $G$-homomorphisms ($G$-equivariant maps);
the dependent sum over the dependent function type
is the quotient of all functions $V_1 \to V_2$ by the conjugation action of $G$.
The invariants (homotopy fixed points) of a $G$-$\infty$-action $\rho$ are the sections of the morphism $V \sslash G \to \mathbf{B}G$,
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
an action as in remark 3, its type of invariants is the dependent product
This is the internal limit in $\mathbf{H}$ of the internal diagram
From def. 2 we read off:
The quotient of a $G$-action
is the dependent sum
This is the internal colimit in $\mathbf{H}$ of the internal diagram
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.
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 $\mathbf{H}$.
Let
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
in $\mathbf{H}$, with the product action being exhibited by the principal ∞-bundle
Here the homotopy fiber on the left is identified as $V_1 \times V_2$ by using that (∞,1)-limits commute over each other.
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 $[V_1, V_2] \in \mathbf{H}$.
Taking fibers
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:
The above internal-hom action
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.
The invariant, def. 3 of the conjugation action, prop. 3 are the action homomorphisms. (See also at Examples - Conjugation actions.)
Therefore
For $\bar \rho_i : V_i \sslash G \to \mathbf{B}G$ two $G$-actions, the object of homomorphisms is
In the syntax of homotopy type theory
See at stabilizer group.
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 $V$
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
which in turn means that the action factors through an action of the stabilizer group $Stab_G(0)$
(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
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
where $\mathbb{D}^V_0$ is the infinitesimal disk around $0$ in $V$.
Here the cartesian subdiagram
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.
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
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
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
Given a discrete group $G$ and an action $\rho$ of $G$ on a set $S$
then the corresponding action groupoid is
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:
For the unique and trivial $G$-action on the singleton set $\ast$, we have
This makes it clear that:
In the situation of def. 6, there is a canonical morphism of groupoids
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.
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. 4
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
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
which strictly preserves the $G$-labels on the morphisms. These are manifestly the intertwiners.
The homotopy fiber of the morphism in prop. 4 is equivalent to the set $S$, regarded as a groupoid with only identity morphisms, hence we have a homotopy fiber sequence of the form
In the presentation $(S//G)_\bullet$ of def. 6, $p_\rho$ is an isofibration, prop. 4. 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.
Given a homotopy fiber sequence of groupoids of the form
such that $S$ is equivalent to a set $S$, define a $G$-action on this set as follows.
Consider the homotopy fiber product
of $i$ with itself. By the pasting law applied to the total homotopy pullback diagram
there is a canonical equivalence of groupoids
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 $\rho$:
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
and hence, via the construction in def. 7, by
But this already exhibits $E$ as an action groupoid, in particular it mans that $\rho$ is really an action:
The morphism $\rho$ constructed in def. 7 is a $G$-action in that it satisfies the action propery, which says that the diagram (of sets)
For $G$ a discrete group, there is an equivalence of categories
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.
By remark 7 the construction of action groupoids is essentially surjective. By prop. 5 it is fully faithful.
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.
Consider the étale geometric morphism
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
The right $\infty$-action of $G$ on itself is given by the fiber sequence
which exhibits $\mathbf{B}G$ as the delooping of $G$.
The fiber sequence
given by the free loop space object $\mathcal{L}\mathbf{B}G$ exhibits the higher adjoint action of $G$ on itself:
For $V \in \mathbf{H}$ any object, there is a canonical action of the internal automorphism infinity-group $\mathbf{Aut}(V)$:
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:
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
(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
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
On the other hand, a morphism of actions
is a function of the underlying sets, such that for all these terms we have
which is equivalent to
Comparison of (1) and (2) shows that the identification
establishes a natural equivalence (a natural bijection of sets in this case)
showing how $[\Sigma,X]$ is indeed the internal hom of $G$-actions.
Generally, for $G$ a discrete ∞-group we have an equivalence of (∞,1)-categories
(by the (∞,1)-Grothendieck construction), and hence
is the (∞,1)-category of ∞-permutation representations.
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. 9
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. 8. Then the internal-hom action of prop. 3
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 $\Sigma$.
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
The corresponding action groupoid $(\mathbf{B}A)\sslash G \simeq \mathbf{B}( G \ltimes A)$ is the delooping of the corresponding semidirect product group.
For $G \in Grp(\mathbf{H})$ the $\infty$-category of $G$-modules is
the stabilization of the $\infty$-category of $G$-actions.
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. 10.
Consider an object $B \in \mathbf{H}$ and an object
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.
By adjunction there is a canonical morphism from the re-pullback of this to the slice automorphism group
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 $L$:
Underlying the $B^\ast\mathbf{Aut}_{\mathbf{H}}(L)$-action on $L$ is an $\mathbf{Aut}_{\mathbf{H}}(L)$-action on
and
Applying $\underset{B}{\sum}$ to the Cartesian diagram that defines the $\infty$-action on $L$
yields
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.
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 $X = \underset{B}{\sum} L$.
Stated more intuitively, prop. 9 says that sliced automorphisms of the form
act on $X$ by the evident restriction to the horizontal equivalences,
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
(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.)
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.
Given an ∞-group $G$ in $\mathbf{H}$ and a $G$-action, def. 2, 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.)
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.
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
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.
See Lie infinity-algebroid representation.
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).
action, $\infty$-action
representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory:
homotopy type theory | representation 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 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 $\mathbf{B}G \to \mathbf{B}H$ | |
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) |
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 $\mathbf{H}= \infty Grpd$ the statement that homotopy types over $B G$ are equivalently $G$-infinity-actions is (via the Borel model structure) due to
This is mentioned for instance as exercise 4.2in
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 $A_\bullet \to B_\bullet$ 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.
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
that
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.