this entry is about the notion of action in algebra (of one algebraic object on another object). For the notion of action functional in physics see there.
symmetric monoidal (∞,1)-category of spectra
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
There are various variants of the notion of something acting on something else. They are all closely related.
For an entity $K$ to be able to act on anything else in the first place, $K$ needs to have some notion of composition or product with itself. Quite generally this is modeled by realizing $K$ as an n-category of sorts. In particular, if $K$ is a monoidal entity with a product $K \otimes K \to K$ there is usually a delooping $\mathbf{B} K$ which encodes the same information.
From this perspective one fundamental way to realize an action of some entity $K$ on some entity in the category or n-category $C$ is an ($n$-)functor
or
Here the collection $\coprod_{k \in Obj(k)} \rho(k)$ of objects in $C$ in the image of objects of $K$ is “what $K$ is acting on” and the images of the morphisms in $K$ under $\rho$ produce the action itself.
An action of a group $G$ on an object $x$ in a category $C$ is a representation of $G$ on $x$, that is a group homomorphism $\rho : G \to Aut(x)$, where $Aut(x)$ is the automorphism group of $x$.
As indicated above, a more sophisticated but equivalent definition treats the group $G$ as a category denoted $\mathbf{B} G$ with one object, say $*$. Then an action of $G$ in the category $C$ is just a functor
Here the object $x$ of the previous definition is just $\rho(*)$.
More generally we can define an action of a monoid $M$ in the category $C$ to be a functor
where $\mathbf{B} M$ is (again) $M$ regarded as a one-object category.
The category of actions of $M$ in $C$ is then defined to be the functor category $C^{\mathbf{B} M}$.
One can also define an action of a category $D$ on the category $C$ as a functor from $C$ to $D$, but usually one just calls this a functor.
Another perspective on the same situation is: a (small) category is a monad in the category of spans in Set. An action of the category is an algebra for this monad. See action of a category on a set.
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 $Cat$ as monoidal bicategory it specializes to the notion of actegory.
Suppose we have a category, $C$, 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, $G$, in $C$ on an object, $X$, of $C$.
By the adjointness relation between cartesian product, $A\times ?$, and function set, $?^A$, in Set, a group homomorphism
corresponds to a function
which will have various properties encoding that $\alpha$ was a homomorphism of groups:
and these can be encoded diagrammatically.
Because of this, we can define an action of a group object, $G$, in $C$ on an object, $X$, of $C$ 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 $C$ 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 symplectic geometry one considers Hamiltonian actions.
(…)
action, ∞-action,