## § Galois correspondence, functorially

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