Take two posets $A, B$ (partially ordered sets). Now consider these posets to be categories $Cat(A), Cat(B)$ respectively. Consider a map from $A$ to $B$, $f: A \rightarrow B$. This can be seen as a functor $F : Cat(A) \rightarrow Cat(B)$. Under what conditions on the posets $A, B$ and the map $f$, will the functor $F$ have an adjoint? I would like to say that if $f$ is Scott-continuous, then $F$ will have an adjoint. Is this true?



a dcpo seen as a category: when does a dcpo map induce a functor with an adjoint?
*I would like to say that if f is Scott-continuous, then F will have an adjoint. Is this true?*


No.


If $A$ and $B$ are complete (semi)lattices then $f:A\to B$ has a right adjoint $f\dashv g$ iff $f$ preserves all joins, and then $g:B\to A$ that preserves all meets, and *vice versa*.


We could call this the **(Special) Adjoint Function/or Theorem**. However, the result is actually so familiar that it is built into the language of mathematics, if not everyday life, for example in phrases like **least common multiple**.


If $A$ and $B$ are finite lattices, every order-preserving function between them is Scott continuous, but $f:A\to B$ has a right adjoint iff it preserves $\bot$ and $\lor$.


Of course, posets or dcpos that are not lattices can have or fail to have adjoints too, but it is really not worth the trouble of formulating (necessarily very complicated) theorems about this.


Contravariant adjunctions between lattices arise *extremely* commonly just from an *arbitrary* binary relation. I feel that this justifies retaining the name *Galois connection* for the contravariant situation.


Let $R\subset X\times Y$ be *any* relation between two sets. For *any* subsets $A\subset X$ and $B\subset Y$, write
$$ A^+ = \{ b | \forall a\in A.R(a,b) \} \text{ and } B^+ = \{ a | \forall b\in B.R(a,b) \}. $$
Then this defines a contravariant adjunction between the powersets of $X$ and $Y$.


In the particular case of Galois theory, let $X$ be the field, $Y$ its automorphism group and $R$ the relation that a particular automorphism fixes a particular element of the field. Then as a *corallary* and not as a necessary hypothesis, $A^+$ is a subgroup for *any subset* $B$ and similarly $B^+$ is a subfield for *any subset* $A$. Under the usual conditions (and only now do we introduce non-trivial algebra) the $(-)^+$ constructions restrict to a bijection between subgroups and subfields.


