Editing Group action (section)

== Variants and generalizations ==
We can also consider actions of [[monoid]]s on sets, by using the same two axioms as above. This does not define bijective maps and equivalence relations however. See [[semigroup action]].

Instead of actions on sets, we can define actions of groups and monoids on objects of an arbitrary category: start with an object {{math|''X''}} of some category, and then define an action on {{math|''X''}} as a monoid homomorphism into the monoid of [[endomorphisms]] of {{math|''X''}}. If {{math|''X''}} has an underlying set, then all definitions and facts stated above can be carried over. For example, if we take the category of vector spaces, we obtain [[group representation]]s in this fashion.

We can view a group {{math|''G''}} as a category with a single object in which every morphism is [[Inverse element|invertible]].<ref>{{harvp|Perrone|2024|pages=7-9}}</ref> A (left) group action is then nothing but a (covariant) [[functor]] from {{math|''G''}} to the [[category of sets]], and a group representation is a functor from {{math|''G''}} to the [[category of vector spaces]].<ref>{{harvp|Perrone|2024|pages=36-39}}</ref> A morphism between {{math|''G''}}-sets is then a [[natural transformation]] between the group action functors.<ref>{{harvp|Perrone|2024|pages=69-71}}</ref>  In analogy, an action of a [[groupoid]] is a functor from the groupoid to the category of sets or to some other category.

In addition to [[continuous group action|continuous actions]] of topological groups on topological spaces, one also often considers [[Lie group action|smooth actions]] of Lie groups on [[manifold|smooth manifold]]s, regular actions of [[algebraic group]]s on [[algebraic variety|algebraic varieties]], and [[group-scheme action|actions]] of [[group scheme]]s on [[scheme (mathematics)|schemes]]. All of these are examples of [[group object]]s acting on objects of their respective category.