Another way to describe -sets is the functor category , where is the groupoid (category) with one element and isomorphic to the group . Indeed, every functor of this category defines a set and for every in (i.e. for every morphism in ) induces a bijection : . The categorical structure of the functor assures us that defines a -action on the set . The (unique) representable functor is the Cayley representation of . In fact, this functor is isomorphic to and so sends to the set which is by definition the "set" and the morphism of (i.e. the element of ) to the permutation of the set . We deduce from the Yoneda embedding that the group is isomorphic to the group , a subgroup of the group of permutations of .
Another way to describe -sets is the functor category , where is the groupoid (category) with one element and isomorphic to the group . Indeed, every functor of this category defines a set and for every in (i.e. for every morphism in ) induces a bijection : . The categorical structure of the functor assures us that defines a -action on the set . The (unique) representable functor is the Cayley representation of . In fact, this functor is isomorphic to and so sends to the set which is by definition the "set" and the morphism of (i.e. the element of ) to the permutation of the set . We deduce from the Yoneda embedding that the group is isomorphic to the group , a subgroup of the group of permutations of .