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 $Y$. 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, which is also possessed by $Y^Y$ and preserved by the curried action $\widehat{act}$. Note that if $Y$ is any set then $Y^Y$ is a monoid, and when $X$ acts on it one calls it an X-set. For $Y^Y$ to have a ring/algebra structure, $Y$ must be some sort of abelian group or vector space with the action by linear functions; then one calls the action also a module or representation.
In terms of the uncurried action $X\times Y\to Y$, the “preservation” condition says roughly speaking that acting consecutively with two elements in $X$ is the same as first multiplying them and then acting with the result:
To be precise, this is the condition for a left action; a right action is defined dually in terms of a map $Y\times X\to Y$. If $X$ has no algebraic structure, or if its relevant structure is commutative, then there is no essential difference between the two; but in general they can be quite different.
This action property can also 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^{1} 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.
The action of a set on a set was defined above; it consists of a function $act: X\times Y\to Y$. This can equivalently be represented by a quiver with $Y$ as its vertices, with its edges labeled by elements of $X$, and such that each vertex has exactly one arrow leaving it with each label. (This is a sort of “Grothendieck construction”.) It is also the same as a simple (non halting) deterministic automaton, with $Y$ the set of states and $X$ the set of inputs.
That an action is a type of edge labeled quiver can be seen by explicitly giving the product projection functions, $p_1$ and $p_2$, of $X\times Y$.
The shape of this diagram corresponds to that of an edge labeled quiver:
While the set $X$ has no algebraic structure to be preserved, the action $act$ generates a unique free category action $act^{*}:X^{*}\times Y\to Y$ where $X^{*}$ is the free monoid on $X$ containing paths of $X$ elements. The monoidal structure of $X^*$ is preserved: two actions in succession is equal to the action of the concatenation of their paths.
A representation is a “linear action”.
In symplectic geometry one considers Hamiltonian actions.
(…)
action, ∞-action,
One example of this relatively rare usage is William Lawvere: Qualitative Distinctions Between Some Toposes of Generalized Graphs, Contemporary Mathematics 92 (1989) in which this sense of action is routinely used. ↩