Semidirect products as group objects II

I kept thinking about the last post and realized a more systematic explanation: We have two kinds of procedures that we can apply to the category of sets, and more generally to any kind of category (that admits finite products). On the one hand, we can form the category of G-sets, which is the functor category [G, Set]. On the other hand, we can form the category of groups, which is the category of group objects in Set. Both of these constructions commute because products in functor categories can be computed pointwise. In this way, we will end up with an equivalence between the functor category [G, Grp], whose objects are the semidirect products of the form (–) ⋊ G, and the category of group objects in the category of G-Sets. Let us be a bit more precise:

We denote the category of groups by Grp and the category of sets by Set. Let 𝒞 be a category admitting finite products, and let us denote by Grp(𝒞) the resulting category of group objects in 𝒞. We have an equivalence of categories between Grp and Grp(Set).

Given another category 𝒢, the resulting functor category [𝒢, 𝒞] again admits finite products. Moreover, these products can be computed pointwise. It follows that we have an equivalence of categories between [𝒢, Grp(𝒞)] and Grp([𝒢, 𝒞]).

We may regard the group G as a one-object groupoid. The category of G-sets, which we denote by G-Set, is then equivalent to the functor category [G, Set].

In my original post, I regarded as “semidirect product of the form (–) ⋊ G” as a pair (H, θ) consisting of a group H and a homomorphism of groups α from G to Aut(H). The automorphism group Aut(H) is precisely the group of automorphisms of H as an object of the category Grp. Such a pair (H, α) is therefore the same as a functor from G to Grp. We can now apply the equivalences of categories [G, Grp] ≃ [G, Grp(Set)] ≃ Grp([G, Set]) ≃ Grp(G-Set) to see that a functor from G to Grp is essentially the same as a group object in the category of G-sets.