I am working though 7 sketches in compositionality and have almost reached the end of chapter 1, which is very much concerned with Galois Connections. One of the questions on the subject that is not yet clear to me is:
Given $A \subseteq B$ and $(A,\leq),(B,\leq)$ both preorders. Lets call $id: A \rightarrow B$ the embedding of $A$ in $B$. In some cases it is clear that there is a right or left adjoint to this embedding (e.g. if $A={\wedge B}$ or $A=B$).
Is there a general way to give conditions for when this embedding allows a Galois connection, and construct the connection when allowed?