M Theory Lesson 220

A simple example of an adjunction between two 1-categories is a Galois connection. We are interested in the case of sets with a partial order, so there exists an arrow from $a$ to $b$ in a set $S$ whenever $a \leq b$. The natural functions (in this case functors) between such sets (here considered as categories) are monotone functions, so a Galois connection consists of two monotone functions $f: S \rightarrow T$ and $g: T \rightarrow S$ such that $f(a) \leq x$ iff $a \leq g(x)$. Observe that, at least when $g$ is onto, $fg$ is idempotent as a map from $T$ to $T$. This follows from $gfg (y) \geq g(y)$ (and $gfg (y) \leq g(y)$), which is true because the application of the one rule to $f(x) \leq f(x)$ gives $x \leq gf (x)$, and we can find a $y$ such that $x = g(y)$.

Usually the sets $S$ and $T$ are quite different. Consider the original example for finite number fields. The finite field with $p^{n}$ elements, for $p$ prime, is an extension of the field with $p$ elements. Take the finite set of all fields in this large field which contain the field with $p$ elements. For any such field $K$, there is a map $K \mapsto \textrm{Gal}(K)$ which sends $K$ to its Galois group in the large field. The dual connection map takes any subgroup $H$ of the main Galois group to all elements of the large field that are fixed by $H$.

Now let $p = 3$ and $n = 3$. There are fields of 3, 9 and 27 elements that extend the field $F_{3}$. The main Galois group is all isomorphisms of the 27 element field that fix the field $F_{3}$. The dual connection map sends the trivial group inside this group to the full 27 element field, the whole Galois group to $F_{3}$ and an intermediary subgroup (bit from trit) to the 9 element field. All this is encoded in an elementary pair of arrows between the 2 kinds of three element set.