§ Galois correspondence, functorially

For a given group GG, build the category of subgroups as follows: The objects aren't exactly subgroups, but are isomorphic to them --- for each coset HH, the category has the object as the coset space G/HG/H, equipped with the left-action of GG on the coset space . The morphisms between G/HG/H and G/KG/K are the intertwining maps ϕ:G/HG/K\phi: G/H \rightarrow G/K which commute with the action of GG: (g×)ϕ=ϕ(g×)(g \times ) \circ \phi = \phi \circ (g \times ). We first work out what it means to have such an intertwining map. Suppose we pick a coset of HH, which is an element of the coset space αHG/H\alpha H \in G/H for some αG\alpha \in G. Now the intertwining condition says that ϕ(gαH)=gϕ(αH)\phi(g \alpha H) = g \phi(\alpha H). If we pick α=e\alpha = e, then we get ϕ(gH)=gϕ(H)\phi(g H) = g \phi(H). Thus, the intertwining map is entirely determined by where it sends HH, ie, the image ϕ(H)\phi(H). Now, let the image of ϕ(H)\phi(H) be some coset γKG/K\gamma K \in G/K. Suppose that the coset gH=gHgH = g'H, since writing a coset as gHgH is not unique. Apply ϕ\phi to both sides and use that ϕ\phi is intertwining. This gives ϕ(gH)=gϕ(H)=gγK\phi(gH) = g \phi(H) = g \gamma K and ϕ(gH)=gϕ(H)=gγK\phi(g'H) = g'\phi(H) = g' \gamma K. For these to be equal, we need γ1g1gγK\gamma^{-1} g'^{-1} g \gamma \in K. But since gH=gHgH = g'H, we know that this is equivalient to gg1Hg g'^{-1} \in H. Thus, the above condition becomes equivalent to hH,γ1hγK\forall h \in H, \gamma^{-1} h \gamma \in K. So we now have a category whose objects are coset spaces and whose morphisms are intertwining maps. We now consider the category of a given field extension L/FL/F, which has objects intermediate fields between LL and FF, and has morphisms as field morphisms which fix FF.