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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
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 subgroup.
Of -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. 7
The slice is the context of types which are generally covariant over .
On consider the trivial -action, def. 6. 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. 8.
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