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.

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Actions

Idea

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 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. The distinction between left and right actions is mirrored in the variance; acting on the left yields a covariant functor, whereas acting on the right is expressed via contravariance.

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.

Definitions

Actions of a group

An action of a group $G$ on an object $S$ in a category $\mathcal{C}$ is a representation of $G$ on $S$, that is a group homomorphism $\rho \colon G \to Aut(S)$, where $Aut(S)$ is the automorphism group of $S$ in $\mathcal{C}$.

Group actions, especially continuous actions on topological spaces, are also known as transformation groups (starting around Klein 1872, Sec. 1, see also Koszul 65 Bredon 72, tom Dieck 79, tom Dieck 87). Alternatively, if the group $G$ that acts is understood, one calls (Bredon 72, Ch. II) the space $X$ equipped with an action by $G$ a topological G-space (or G-set, G-manifold, etc., as the case may be).

As indicated above, a more abstract but equivalent definition regards the group $G$ as a category (a groupoid), denoted $\mathbf{B} G$, with a single object $\ast$. Then an action of $G$ in the category $C$ is equivalently a functor of the form

Here the object $S$ of the previous definition is the $\rho(\ast)$ of that single object.

Concretely, if $\mathcal{C}$ is a category like Set, then an action is equivalently a function

which satisfies the action property

and

Actions of a monoid

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 $C^{\mathbf{B} M}$. When $C$ is $Set$ this is called MSet.

Considering this in enriched category theory yields the internal notion of action objects.

Actions of a category

One can¹ 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 module category (also “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 module category.

Actions of a group object

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, only locally compact topological spaces have well-behaved topological automorphism groups, and thus actions of topological spaces on topological spaces must either be restricted to actions on locally compact spaces, or else be defined as above.

As another example, within the category of profinite groups viewed as topological groups, not all objects have automorphism groups for which the natural topology is profinite. Thus profinite group actions on (the underlying topological space of) a profinite group must either be given in this form, or else be restricted to actions on profinite groups for which the automorphism group is naturally profinite.

Actions of a monoid object

Suppose we have a category, $C$, with binary products and a terminal object $*$. There is an alternative way of viewing monoid actions in Set, so that we can talk about an action of a monoid object, $M$, in $C$ on an object, $X$, of $C$.

By the adjointness relation between cartesian product, $A\times ?$, and function set, $?^A$, in Set, a monoid homomorphism

corresponds to a function

which will have various properties encoding that $\alpha$ was a homomorphism of monoids:

and these can be encoded diagrammatically.

Because of this, we can define an action of a monoid object, $M$, 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 endomorphism monoid objects for all objects being considered.

Actions of a set

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.

An action of a set $X$ in itself is also called a binary operation, and the set $X$ is called a magma.

Examples

• A representation is a “linear action”.

• In symplectic geometry one considers Hamiltonian actions.

• In topology: topological G-space

  • circle action

  • group action on an n-sphere

(…)

Related concepts

• action object

• action, ∞-action,

  • conjugation action, adjoint action

  • diagonal action

  • transitive action, free action, regular action

  • proper action, properly discontinuous action

• coaction

• representation, ∞-representation

• module, ∞-module

• action monad

• associated bundle, associated ∞-bundle

• quotient, quotient stack, quotient type

• representation theory, invariant theory

• equivariant homotopy theory

  • Borel model structure

• biaction

References

Group actions

On group actions, mostly in TopologicalSpaces, hence in the form of topological G-spaces:

Historical origins:

• Felix Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen (1872) Mathematische Annalen volume 43, pages 63–100 1893 (doi:10.1007/BF01446615)

  English translation by M. W. Haskell:

  A comparative review of recent researches in geometry, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (euclid:1183407629, LaTeX version retyped by Nitin C. Rughoonauth: arXiv:0807.3161)

Introduction of group actions into (quantum) physics (cf. Gruppenpest):

• Hermann Weyl, §III.1 in: Gruppentheorie und Quantenmechanik, S. Hirzel, Leipzig (1931), translated by H. P. Robertson: The Theory of Groups and Quantum Mechanics, Dover (1950) [ISBN:0486602699, ark:/13960/t1kh1w36w]

Textbook accounts:

• Glen Bredon, Introduction to compact transformation groups, Academic Press 1972 (ISBN 9780080873596

  , pdf)

• Tammo tom Dieck, Transformation Groups and Representation Theory, Lecture Notes in Mathematics 766, Springer 1979 (doi:10.1007/BFb0085965)

• Tammo tom Dieck, Transformation Groups, de Gruyter 1987 (doi:10.1515/9783110858372)

Lecture notes:

• Jean-Louis Koszul, Lectures on Groups of Transformations, Tata Institute 1965 (pdf, pdf)

• Patrick Morandi, Group actions (pdf)