Semidirect products as group objects I

Yesterday, I was watching the talk “A Short Introduction to Categorical Logic” by Evan Patterson on YouTube. Around the midpoint of the talk, some examples for group objects are mentioned. I was aware of the first three examples (groups, topological groups, and Lie groups), but I hadn’t seen the last example before: apparently, given a group G, group objects in the category of G-sets are (in some sense) the same semidirect products of the form (–) ⋊ G.

Let us check this.

A group object in the category of G-sets is — by definition — a G-set H together with a G-equivariant multiplication map μ : H × H → H, a G-equivariant inversion map ι : H → H, and a G-equivariant map {★} → H picking out an element e of H, such that these maps make H into a group. (More precisely, the map μ is the multiplication on H, the map ι assigns to each element of H its inverse, and the element e is the neutral element for the group structure.)

The forgetful functor from the category of G-sets to the category of sets preserves products, and therefore preserves group objects. The group object H is therefore the same as a group that is also a G-set and for which

• we have g . (h₁ · h₂) = (g . h₁) · (g . h₂),
• we have g . e = e, and
• we have (g . h)⁻¹ = g . h⁻¹.

The first condition tells us that G needs to act on H via group automorphisms, and the second and third condition follows from this.

We hence find that a group object in the category of G-sets is the same as a group H together with a homorphism of groups from G to Aut(H). This is, in turn, the same as a semidirect product of the form H ⋊ G.