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 an entity to be able to act on anything else in the first place, needs to have some notion of composition or product with itself. Quite generally this is modeled by realizing as an n-category of sorts. In particular, if is a monoidal entity with a product there is usually a delooping which encodes the same information.
Here the collection of objects in in the image of objects of is “what is acting on” and the images of the morphisms in under produce the action itself.
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.