Free Actions, Ch III: Group Actions

This lecture introduces the concept of a free action functor, which is the left adjunct of the forgotten functor from the category of G-sets to sets. The construction of free actions is explained, emphasizing the absence of constraints in the action. The proof that the free action functor is indeed the left adjunct of the forgotten functor is outlined. The lecture also covers the preservation of identities and compositions by the free action functor, along with the natural bijection it exhibits. The g-equivariant nature of the free action on GxX is demonstrated, and the lecture concludes with the establishment of the naturalness of the application through precomposition and a simple calculation.