Algebras and modules
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The notion of -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 may act on another object by a morphism which satisfies an action property up to coherent higher homotopy.
If the -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 be an (∞,1)-topos.
Let be an group object in an (∞,1)-category in , hence a homotopy-simplicial object on of the form
satisfying the groupoidal Segal conditions.
hence an ∞-group.
An action (or -action, for emphasis) of on an object 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 .
The (∞,1)-category of such actions is the slice of groupoid objects over 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 .
Every -action has a classifying morphism in that there is a fiber sequence
such that is the -action on regarded as the corresponding -principal ∞-bundle modulated by .
This allows to characterize -actions in the following convenient way. See (NSS) for a detailed discussion.
For an object, a --action on is a fiber sequence in of the form
The (∞,1)-category of -actions in is the slice (∞,1)-topos of over :
Notions in higher representation theory
We discuss some basic representation theoretic notions of -actions.
In summary, for an action of on , we have
And for two actions we have
Coinvariants / Quotients
From def. 2 we read off:
The quotient of a -action
is the dependent sum
We describe here aspects of the cartesian product and internal hom of -actions given this way. The following statements are essentially immediate consequences of basic homotopy type theory.
For their cartesian product is a -action on the product of with in .
be the principal ∞-bundles exhibiting the two actions.
Along the lines of the discussion at locally cartesian closed category we find that is given in by the (∞,1)-pullback
in , with the product action being exhibited by the principal ∞-bundle
Here the homotopy fiber on the left is identified as by using that (∞,1)-limits commute over each other.
For their internal hom is a -action on the internal hom .
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:
Internal object of homomorphisms
For two -actions, the object of homomorphisms is
In the syntax of homotopy type theory
See at stabilizer group.
We discuss linearization of -actions using the axioms of differential cohesion.
Let be a pointed object.
Let be an -group acting on
such that this action preserves the point of , 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
(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 -action on restricts to the trivial action on the point of .
Now let denote the infinitesimal shape modality. Since it preserves the top homotopy pullback, it follows that applying the orthogonal factorization system (-equivalences, formally etale morphisms) to the top vertical morphisms produces a pasting diagram of homotopy pullbacks of the form
where is the infinitesimal disk around in .
Here the cartesian subdiagram
hence exhibits a -action on .
Any -action on an infinitesimal disk is a linear action, given by a homomorphism to the automorphism infinity-group of the infinitesimal disk, the general linear group of the tangent space of at 0.
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.
for a groupoid object given by an explicit choice of set of objects and of morphisms and then write for the object that this presents in the -category. Given any such , we recover a presentation by choosing any essentially surjective functor (an atlas) out of a set (regarded as a groupoid) and setting
hence taking as the set of objects and the homotopy fiber product of with itself over as the set of morphism.
For a discrete group, then denotes the groupoid presented by 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 and an action of on a set
then the corresponding action groupoid is
with composition given by the product in . Hence the objects of are the elements of , and the morphisms are labeled by elements and are such that .
For the unique and trivial -action on the singleton set , 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 a discrete group, given two -actions and on sets and , respectively, then there is a natural equivalence between the set of action homomorphisms (“intertwiners”) , regarded as a groupoid with only identity morphisms, and the hom groupoid of the slice 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 is an isofibration, so is , 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 -labels on the morphisms. These are manifestly the intertwiners.
The homotopy fiber of the morphism in prop. 4 is equivalent to the set , regarded as a groupoid with only identity morphisms, hence we have a homotopy fiber sequence of the form
In the presentation of def. 6, is an isofibration, prop. 4. Hence the homotopy fibers of are equivalent to the ordinary fibers of computed in the 1-category of 1-groupoids. Since is the identity on the labels of the morphisms in this presentation, this ordinary fiber is precisely the sub-groupoid of consisting of only the identity morphismss, hence is the set 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 is equivalent to a set , define a -action on this set as follows.
Consider the homotopy fiber product
of 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 is projection on the first factor. The other map under this equivalence we denote by :
But this already exhibits as an action groupoid, in particular it mans that is really an action:
The morphism constructed in def. 7 is a -action in that it satisfies the action propery, which says that the diagram (of sets)
-group actions in an -topos
Let be an (∞,1)-topos and let be an ∞-group in .
The following lists some fundamental classes of examples of -actions of , and of other canonical -groups. By the discussion above these actions may be given by the classifying morphisms.
Consider the étale geometric morphism
For any object, the trivial action of on is , exhibited by the split fiber sequence
The right -action of on itself is given by the fiber sequence
which exhibits as the delooping of .
The fiber sequence
given by the free loop space object exhibits the higher adjoint action of on itself:
For any object, there is a canonical action of the internal automorphism infinity-group :
We discuss the simple case of the cartesian closed category of -sets (G-permutation representations) for an ordinary discrete group as a simple illustration of the internal hom of -actions, prop. 3.
This example spells out everything completely in components:
Let ∞Grpd, let be an ordinary discrete group and let be sets equipped with -action (permutation representations).
In this case is simply the set of functions of sets. Its -action as the internal hom of -actions given, for every and , by
(where we write generically for the given action on the set specified implicitly by the type of the argument).
Hence a morphism of -actions
is a function of the underlying sets such that for all , and all 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 is indeed the internal hom of -actions.
Let be a moduli infinity-stack for field in a gauge theory or sigma-model. Let be the corresponding spacetime or worldvolume, respectively.
We have the automorphism action, def. 9
The slice is the context of types which are generally covariant over .
On consider the trivial -action, def. 8. Then the internal-hom action of prop. 3
is the configuration space of fields on modulo automorphisms (diffeomorphisms, in smooth cohesion) of . This is the configuration space of “generally covariant” field theory on .
Semidirect product groups
Let be 0-truncated group objects and let be an action of on by group homomorphisms. This is equivalently an action of on , hence a fiber sequence
The corresponding action groupoid is the delooping of the corresponding semidirect product group.
For the -category of -modules is
the stabilization of the -category of -actions.
For and 0-truncated groups, an abelian group with -module structure, the semidirect product group from above exhibits as a -module in the sense of def. 10.
Actions in a slice
Consider an object and an object
in the slice. By the discussion of conjugation actions above, the automorphism ∞-group of as an object in is the dependent product over the automorphism ∞-group in the slice.
By adjunction there is a canonical morphism from the re-pullback of this to the slice automorphism group
Hence the canonical -action on in the slice pulls back to give an action of on :
Underlying the -action on is an -action on
Applying to the Cartesian diagram that defines the -action on
which is still Cartesian, by this proposition. Use that the bottom left object here is equivalently and form the pasting with the naturality square of the -counit.
By this proposition also this naturality square is Cartesian. Hence by the pasting law the total rectangle is Cartesian. This exhibits the -action on .
Co-Discretization of Actions
Let be a local (∞,1)-topos (for instance a cohesive (∞,1)-topos) and write for its sharp modality. Write for the n-image of itd unit.
Given an ∞-group in and a -action, def. 2, on some , then is itself canonically an -group equipped with a canonically induced action on such that the projection carries the structure of a homomorphism of -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 preserves products, since does (being a right adjoint) and by this proposition. Now use that the homotopy quotient is the realization of the simplicial object . So applying to this yields a simplicial object which exhibits the desired action.
Generally, let be any dependent type family (speaking homotopy type theory). We claim that there is an induced family such that for any , where is the inclusion. Applying this when is and when is (necessarily) the basepoint of gives the desired action on the desired type.
First of all, we have the composite , where . Since is itself (since is lex), this factors through , giving a type family such that for any , where is the unit of .
Now fix and . For any and , we can define the type . This is an -type, and since the type of truncated types is an -type, as a function of , this construction factors through . Thus, for and and we have a type , such that
Now by definition, . Thus, we can define by . And since , we have , which is by definition.
Infinitesimally: actions of -algebroids
See Lie infinity-algebroid representation.
Model category presentation
In the context of geometrically discrete ∞-groupoids a model category structure presenting the (∞,1)-category of -actions is the Borel model structure (DDK 80).
representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory:
Actions of A-∞ algebras in some symmetric monoidal (∞,1)-category are discussed in section 4.2 of
Aspects of actions of ∞-groups in an ∞-topos in the contect of associated ∞-bundles are discussed in section I 4.1 of
For discrete geometry
For the statement that homotopy types over are equivalently -infinity-actions is (via the Borel model structure) due to
- E. Dror, William Dwyer, and Daniel Kan, Equivariant maps which are self homotopy equivalences, Proc. Amer. Math. Soc. 80 (1980), no. 4, 670–672 (JSTOR)
This is mentioned for instance as exercise 4.2in
- William Dwyer, Homotopy theory of classifying spaces, Lecture notes Copenhagen (June, 2008) pdf
Closely related discussion of homotopy fiber sequences and homotopy action but in terms of Segal spaces is in section 5 of
There, conditions are given for a morphism to a reduced Segal space to have a fixed homotopy fiber, and hence encode an action of the loop group of on that fiber.
For actions of topological groups
That -actions for a topological group in the sense of G-spaces in equivariant homotopy theory (and hence with not regarded as the geometrically discrete ∞-group of its underying homotopy type ) are equivalently objects in the slice (∞,1)-topos over is Elmendorf's theorem together with the fact, highlighted in this context in
is therefore the slice of the -topos over the global orbit category by .
Rezk-global equivariant homotopy theory:
See at equivariant homotopy theory for more references along these lines.