§ Galois correspondence, functorially
For a given group , build the category of subgroups as follows:
The objects aren't exactly subgroups, but are isomorphic to them ---
for each coset , the category has the object as the coset space ,
equipped with the left-action of on the coset space . The morphisms
between and are the intertwining maps
which commute with the action of : .
We first work out what it means to have such an intertwining map. Suppose we pick
a coset of , which is an element of the coset space for some .
Now the intertwining condition says that . If we pick
, then we get . Thus, the intertwining map
is entirely determined by where it sends , ie, the image .
Now, let the image of be some coset . Suppose
that the coset , since writing a coset as is not unique. Apply
to both sides and use that is intertwining. This gives
For these to be equal, we need . But since
, we know that this is equivalient to . Thus, the above condition becomes
equivalent to .
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 , which has objects intermediate fields
between and , and has morphisms as field morphisms which fix .