symmetric monoidal (∞,1)-category of spectra
There are various variants of the notion of something acting on something else. They are all closely related.
For fixed this produces an endofunction and hence some “transformation” or “action” on . In this way the whole of acts on .
Usually the key aspect of an action of some is that itself carries an algebraic structure, such as being a group (or just a monoid) or being a ring or an associative algebra. (If these structures act on an abelian group or vector space by linear functions, then one calls the action also a module or representation.)
With such algebraic structure on , the action crucially is to respect this structure in that acting consecutively with two elements in is the same as first multiplying them and then acting with the result:
In this way essentially every kind of functor, n-functor and enriched functor may be thought of as defining a generalized kind of action. This perspective on actions is particularly prevalent in enriched category theory, where for instance coends may be thought of as producing tensor products of actions in this general functorial sense.
Here the total space of this bundle is typically the “weak” quotient (for instance: homotopy quotient) of the action, whence the notation. If one thinks of as the classifying space for the -universal principal bundle, then this bundle is the -fiber bundle which is associated via the action to this universal bundle. For more on this perspective on actions see at ∞-action.
As indicated above, a more sophisticated but equivalent definition treats the group as a category denoted with one object, say . Then an action of in the category is just a functor
Here the object of the previous definition is just .
More generally we can define an action of a monoid in the category to be a functor
where is (again) regarded as a one-object category.
The category of actions of in is then defined to be the functor category .
On the other hand, an action of a monoidal category (not in a monoidal category, as above) is called an actegory. This notion can be expanded of course to actions in a monoidal bicategory, where in the case of as monoidal bicategory it specializes to the notion of actegory.
Suppose we have a category, , with binary products and a terminal object . There is an alternative way of viewing group actions in Set, so that we can talk about an action of a group object, , in on an object, , of .
By the adjointness relation between cartesian product, , and function set, , in Set, a group homomorphism
corresponds to a function
which will have various properties encoding that was a homomorphism of groups:
and these can be encoded diagrammatically.
Because of this, we can define an action of a group object, , in on an object, , of to be a morphism
satisfying conditions that certain diagrams (left to the reader) encoding these two rules, commute.
The advantage of this is that it does not require the category to have internal automorphism group objects for all objects being considered.
As an example, within the category of profinite groups, not all objects have automorphism groups for which the natural topology is profinite, because of that profinite group actions are sometimes given in this form or are restricted to actions on objects for which the automorphism group is naturally profinite.
A representation is a “linear action”.
In the category Set there is no difference between the above left action and the right action because the cartesian product is symmetric monoidal. However for the action of a monoid on a set (sometimes called M-set or M-act) the product of a monoid and a set does not commute so the left and right actions are different. The action of a set on a set is the same as an arrow labeled directed graph which specifies that each vertex must have a set of arrows leaving it with one arrow per label, and is also the same as a simple (non halting) deterministic automaton . ↩