nLab
action

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.


Context

Algebra

Representation theory

Actions

Idea

There are various variants of the notion of something acting on something else. They are all closely related.

For an entity KK to be able to act on anything else in the first place, KK needs to have some notion of composition or product with itself. Quite generally this is modeled by realizing KK as an n-category of sorts. In particular, if KK is a monoidal entity with a product KKKK \otimes K \to K there is usually a delooping BK\mathbf{B} K which encodes the same information.

From this perspective one fundamental way to realize an action of some entity KK on some entity in the category or n-category CC is an (nn-)functor

ρ:KC \rho : K \to C

or

ρ:BKC \rho : \mathbf{B} K \to C

Here the collection kObj(k)ρ(k)\coprod_{k \in Obj(k)} \rho(k) of objects in CC in the image of objects of KK is “what KK is acting on” and the images of the morphisms in KK under ρ\rho produce the action itself.

Definitions

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 CC to DD, 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.

Examples

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References

Revised on April 15, 2014 03:36:33 by Urs Schreiber (185.37.147.12)