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.



Representation theory



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, XX, acting on another set YY and such an action is given by a function from the product of XX with YY to YY

act:X×YY. act\colon X \times Y \to Y \,.

For fixed xXx \in X this produces an endofunction act(x,):YYact(x,-) \colon Y \to Y and hence some “transformation” or “action” on YY. In this way the whole of XX acts on YY.

Here (xact(x,))(x\mapsto act(x,-)) is the curried function act^:XY Y\widehat{act}\colon X \to Y^Y of the original actact, which maps XX to the set of endofunctions on YY. Quite generally one has these two perspectives on actions.

Usually the key aspect of an action of some XX is that XX 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 YY^Y and preserved by the curried action act^\widehat{act}. Note that if YY is any set then Y YY^Y is a monoid, and when XX acts on it one calls it an X-set. For Y YY^Y to have a ring/algebra structure, YY 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×YYX\times Y\to Y, the “preservation” condition says roughly speaking that acting consecutively with two elements in XX is the same as first multiplying them and then acting with the result:

act(x 2,act(x 1,y))=act(x 2x 1,y). act(x_2,act(x_1,y)) = act(x_2\cdot x_1, y) \,.

To be precise, this is the condition for a left action; a right action is defined dually in terms of a map Y×XYY\times X\to Y. If XX 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 BX\mathbf{B}X of the monoid XX to the category (such as Set) of which YY 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 XX is some bundle Y/XY/X over BX\mathbf{B}X, whose fiber is YY:

Y Y/X BX. \array{ Y &\longrightarrow& Y/X \\ && \downarrow \\ && \mathbf{B}X } \,.

Here the total space Y/XY/X of this bundle is typically the “weak” quotient (for instance: homotopy quotient) of the action, whence the notation. If one thinks of BX\mathbf{B}X as the classifying space for the XX-universal principal bundle, then this bundle Y/XBXY/X \to \mathbf{B}X is the YY-fiber bundle which is associated via the action to this universal bundle. For more on this perspective on actions see at ∞-action.


Actions of a group

An action of a group GG on an object xx in a category CC is a representation of GG on xx, that is a group homomorphism ρ:GAut(x)\rho : G \to Aut(x), where Aut(x)Aut(x) is the automorphism group of xx.

As indicated above, a more sophisticated but equivalent definition treats the group GG as a category denoted BG\mathbf{B} G with one object, say **. Then an action of GG in the category CC is just a functor

ρ:BGC.\rho : \mathbf{B} G \to C.

Here the object xx of the previous definition is just ρ(*)\rho(*).

Actions of a monoid

More generally we can define an action of a monoid MM in the category CC to be a functor

ρ:BMC\rho: \mathbf{B} M \to C

where BM\mathbf{B} M is (again) MM regarded as a one-object category.

The category of actions of MM in CC is then defined to be the functor category C BMC^{\mathbf{B} M}. When CC is SetSet this is called MSet.

Actions of a category

One can1 also define an action of a category DD on the category CC as a functor from DD to CC, 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 CatCat as monoidal bicategory it specializes to the notion of actegory.

Actions of a group object

Suppose we have a category, CC, 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, GG, in CC on an object, XX, of CC.

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

α:GAut(X)\alpha: G\to Aut(X)

corresponds to a function

act:G×XXact: G\times X\to X

which will have various properties encoding that α\alpha was a homomorphism of groups:

act(g 1g 2,x)=act(g 1,act(g 2,x))act(g_1g_2,x) = act(g_1,act(g_2,x))
act(1,x)=xact(1,x) = x

and these can be encoded diagrammatically.

Because of this, we can define an action of a group object, GG, in CC on an object, XX, of CC to be a morphism

act:G×XXact: G\times X\to X

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 CC 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 profinite group (on a set)

See action of a profinite group?.

Actions of a set

The action of a set on a set was defined above; it consists of a function act:X×YYact: X\times Y\to Y. This can equivalently be represented by a quiver with YY as its vertices, with its edges labeled by elements of XX, 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 YY the set of states and XX 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 1p_1 and p 2p_2, of X×YX\times Y.

Xp 1X×Yactp 2YX\overset{\quad p_1 \quad}{ \leftarrow}X\times Y\underoverset{\quad act \quad}{p_2}{\rightrightarrows}Y

The shape of this diagram corresponds to that of an edge labeled quiver:

LabelslabelEdgestargetsourceVerticesLabels\overset{\quad label \quad}{ \leftarrow}Edges\underoverset{\quad target \quad}{source}{\rightrightarrows}Vertices

While the set XX has no algebraic structure to be preserved, the action actact generates a unique free category action act *:X *×YYact^{*}:X^{*}\times Y\to Y where X *X^{*} is the free monoid on XX containing paths of XX elements. The monoidal structure of X *X^* is preserved: two actions in succession is equal to the action of the concatenation of their paths.

act *(x 2 *,act(x 1 *,y))=act *(x 2 *x 1 *,y)act^{*}(x^{*}_2,act(x^{*}_1,y)) = act^{*}(x^{*}_2\cdot x^{*}_1, y)



Related concepts


  1. 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.

Last revised on June 26, 2020 at 10:23:12. See the history of this page for a list of all contributions to it.