TITLE: Upgrade adjunction to equivalence
QUESTION [10 upvotes]: I'm studying category theory by myself and I just came across this sentence from Wikipedia:
An adjunction between categories C and D is somewhat akin to a "weak form" of an equivalence between C and D, and indeed every equivalence is an adjunction. In many situations, an adjunction can be "upgraded" to an equivalence, by a suitable natural modification of the involved categories and functors.
Can someone provide an example of an "upgrade" of an adjunction to an equivalence? I'm interested in understanding why I could intuitively think of an adjunction as a weak form of an equivalence.

REPLY [9 votes]: I think the author of the wikipedia article probably had in mind Leonid Positselski's first answer, where one restricts to the full subcategory of fixed points of the adjunction. Beware there is no guarantee that the fixed points are nonempty! For example, if $F: Set^\to_\leftarrow Ab: U$ is the free/forgetful adjunction bewteen sets and abelian groups, the fixed points are empty.
Here's an illustrative example to have in mind which is not so degenerate. Let $K/k$ be a Galois extension. Then there is an adjuntion between the poset of intermediate subfields $k \subseteq L \subseteq K$ and the opposite of the poset of subgroups of of $Gal(K/k)$; in one direction we send a group to its field of fixed points and in the other direction we send a field to the group of automorphisms that fix it.
This adjunction is typically not an equivalence, but we can pass to the fixed points of the adjunction to get an equivalence between the poset of normal subgroups of $Gal(K/k)$ and the opposite of the poset of intermediate Galois extensions $k \subseteq L \subseteq K$.
Thus the fundamental theorem of Galois theory may be viewed as calculating the fixed point set of an adjunction, and thus as identifying where an adjunction restricts to an equivalence.

REPLY [7 votes]: I agree that Leonid Positselski’s first answer seems probably what the writer had in mind: given an adjunction, restricting to the categories of “fixed points” on each side yields an equivalence.  Here are two important examples in nature, both involving the category of topological spaces:

There’s an adjunction between the categories of preordered sets and topological spaces, sending a preordered set $(X,\leq)$ to $X$ with the topology of down-closed sets, and sending a topological space $Y$ to $Y$ with its specialisation order.  All preorders are fixpoints; on the other side, the fixpoints are exactly the Alexandrov spaces, i.e. spaces where arbitrary intersections of opens are open.  Restricting to this subcategory shows that the category of preorders is equivalent (in fact, isomorphic!) to the category of Alexandrov spaces.
The adjunction between the categories of topological spaces and locales, sending a topological space to its frame/locale of opens and sending a locale to its space of points, restricts to the equivalence between spatial locales and sober spaces.