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.
The simplest concept of an action involves one set, $X$, acting on another set $Y$ and such an action is given by a function from the product of $X$ with $Y$ to $Y$
For fixed $x \in X$ this produces an endofunction $act(x,-) \colon Y \to Y$ and hence some “transformation” or “action” on $Y$. In this way the whole of $X$ acts on $Y$.
Here $(x\mapsto act(x,-))$ is the curried function $\widehat{act}\colon X \to Y^Y$ of the original $act$, which maps $X$ to the set of of endofunctions on $Y$.^{1} Quite generally one has these two perspectives on actions.
Usually the key aspect of an action of some $X$ is that $X$ 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 $Y$ by linear functions, then one calls the action also a module or representation.)
With such algebraic structure on $X$, the action crucially is to respect this structure in that acting consecutively with two elements in $X$ is the same as first multiplying them and then acting with the result:
This action property may often be identified with a functor property: it characterizes a functor from the delooping $\mathbf{B}X$ of the monoid $X$ to the category (such as Set) of which $Y$ is an object.
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.
Under the Grothendieck construction (or one of its variants), this perspective turns into the perspective where an action of $X$ is some bundle $Y/X$ over $\mathbf{B}X$, whose fiber is $Y$:
Here the total space $Y/X$ of this bundle is typically the “weak” quotient (for instance: homotopy quotient) of the action, whence the notation. If one thinks of $\mathbf{B}X$ as the classifying space for the $X$-universal principal bundle, then this bundle $Y/X \to \mathbf{B}X$ is the $Y$-fiber bundle which is associated via the action to this universal bundle. For more on this perspective on actions see at ∞-action.
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 $D$ to $C$, 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,
In the category Set there is no difference between the above left action and the right action $actR\colon Y \times X \to Y$ 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 $arrows\colon vertices \times labels \to vertices$ 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 $transition\colon inputs \times states \to states$. ↩