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 of endofunctions on YY.1 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. (If these structures act on an abelian group or vector space YY by linear functions, then one calls the action also a module or representation.)

With such algebraic structure on XX, the action crucially is to respect this structure in 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) \,.

This action property may 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.

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}.

Actions of a category

One can 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, 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.



Related concepts


  1. In the category Set there is no difference between the above left action and the right action actR:Y×XYactR\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:vertices×labelsverticesarrows\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:inputs×statesstatestransition\colon inputs \times states \to states.

Revised on February 2, 2017 09:40:43 by Urs Schreiber (