Galois connection

The notion of a Galois connection has its root in Galois theory. By the fundamental theorem of Galois theory (http://planetmath.org/FundamentalTheoremOfGaloisTheory), there is a one-to-one correspondence between the intermediate fields between a field $L$ and its subfield $F$ (with appropriate conditions imposed on the extension $L/F$), and the subgroups of the Galois group $\mathrm{Gal}(L/F)$ such that the bijection is inclusion-reversing:

$$\mathrm{Gal}(L/F)\supseteq H\supseteq ⟨e⟩\mathit{\hspace{1em}}\text{ iff }\mathit{\hspace{1em}}F\subseteq L^H\subseteq L,\text{ and}$$

$$F\subseteq K\subseteq L\mathit{\hspace{1em}}\text{ iff }\mathit{\hspace{1em}}\mathrm{Gal}(L/F)\supseteq \mathrm{Gal}(L/K)\supseteq ⟨e⟩.$$

If the language of Galois theory is distilled from the above paragraph, what remains reduces to a more basic and general concept in the theory of ordered-sets:

Definition. Let $(P,{\le }_P)$ and $(Q,{\le }_Q)$ be two posets. A Galois connection between $(P,{\le }_P)$ and $(Q,{\le }_Q)$ is a pair of functions $f:=(f^{*},f_{*})$ with $f^{*}:P\to Q$ and $f_{*}:Q\to P$, such that, for all $p\in P$ and $q\in Q$, we have

$$f^{*}(p){\le }_{Q}q\mathit{\hspace{1em}}\text{ iff }\mathit{\hspace{1em}}p{\le }_P{f}_{*}(q).$$

We denote a Galoisconnection between $P$ and $Q$ by $P\stackrel{f}{⊸}Q$, or simply $P⊸Q$.

If we define ${\le }_P^{\prime }$ on $P$ by $a{\le }_P^{\prime }b$ iff $b{\le }_{P}a$, and define ${\le }_Q^{\prime }$ on $Q$ by $c{\le }_Q^{\prime }d$ iff $d{\le }_{Q}c$, then $(P,{\le }_P^{\prime })$ and $(Q,{\le }_Q^{\prime })$ are posets, (the duals of $(P,{\le }_P)$ and $(Q,{\le }_Q)$). The existence of a Galois connection between $(P,{\le }_P)$ and $(Q,{\le }_Q)$ is the same as the existence of a Galois connection between $(Q,{\le }_Q^{\prime })$ and $(P,{\le }_P^{\prime })$. In short, we say that there is a Galois connection between $P$ and $Q$ if there is a Galois connection between two posets $S$ and $T$ where $P$ and $Q$ are the underlying sets (of $S$ and $T$ respectively). With this, we may say without confusion that “a Galois connection exists between $P$ and $Q$ iff a Galois connection exists between $Q$ and $P$”.

Remarks.

1. Since $f^{*}(p){\le }_Q{f}^{*}(p)$ for all $p\in P$, then by definition, $p{\le }_P{f}_{*}{f}^{*}(p)$. Alternatively, we can write

   $1_P{\le }_P{f}_{*}{f}^{*},$(1)

   where $1_P$ stands for the identity map on $P$. Similarly, if $1_Q$ is the identity map on $Q$, then

   $f^{*}{f}_{*}{\le }_Q{1}_Q.$(2)

2. Suppose $a{\le }_{P}b$. Since $b{\le }_P{f}_{*}{f}^{*}(b)$ by the remark above, $a{\le }_P{f}_{*}{f}^{*}(b)$ and so by definition, $f^{*}(a){\le }_Q{f}^{*}(b)$. This shows that $f^{*}$ is monotone. Likewise, $f_{*}$ is also monotone.

3. Now back to Inequality (1), $1_P{\le }_P{f}_{*}{f}^{*}$ in the first remark. Applying the second remark, we obtain

   $f^{*}{\le }_Q{f}^{*}{f}_{*}{f}^{*}.$(3)

   Next, according to Inequality (2), $f^{*}{f}_{*}(q){\le }_{Q}q$ for any $q\in Q$, it is true, in particular, when $q=f^{*}(p)$. Therefore, we also have

   $f^{*}{f}_{*}{f}^{*}{\le }_Q{f}^{*}.$(4)

   Putting Inequalities (3) and (4) together we have

   $f^{*}{f}_{*}{f}^{*}=f^{*}.$(5)

   Similarly,

   $f_{*}{f}^{*}{f}_{*}=f_{*}.$(6)

4. If $(f,g)$ and $(f,h)$ are Galois connections between $(P,{\le }_P)$ and $(Q,{\le }_Q)$, then $g=h$. To see this, observe that $p{\le }_{P}g(q)$ iff $f(p){\le }_{Q}q$ iff $p{\le }_{P}h(q)$, for any $p\in P$ and $q\in Q$. In particular, setting $p=g(q)$, we get $g(q){\le }_{P}h(q)$ since $g(q){\le }_{P}g(q)$. Similarly, $h(q){\le }_{P}g(q)$, and therefore $g=h$. By a similarly argument, if $(g,f)$ and $(h,f)$ are Galois connections between $(P,{\le }_P)$ and $(Q,{\le }_Q)$, then $g=h$. Because of this uniqueness property, in a Galois connection $f=(f^{*},f_{*})$, $f^{*}$ is called the upper adjoint of $f_{*}$ and $f_{*}$ the lower adjoint of $f^{*}$.

Examples.

• The most famous example is already mentioned in the first paragraph above: let $L$ is a finite-dimensional Galois extension of a field $F$, and $G:=\mathrm{Gal}(L/F)$ is the Galois group of $L$ over $F$. If we define

  • $P=\{K\mid K\text{ is a field such that }F\subseteq K\subseteq L\},$ with ${\le }_P=\subseteq$,

  • $Q=\{H\mid H\text{ is a subgroup of }G\},$ with ${\le }_Q=\supseteq$,

  • $f^{*}:P\to Q$ by $f^{*}(K)=\mathrm{Gal}(L/K)$, and

  • $f_{*}:Q\to P$ by $f_{*}(H)=L^H$, the fixed field of $H$ in $L$.

  Then, by the fundamental theorem of Galois theory, $f^{*}$ and $f_{*}$ are bijections, and $(f^{*},f_{*})$ is a Galois connection between $P$ and $Q$.

• Let $X$ be a topological space. Define $P$ be the set of all open subsets of $X$ and $Q$ the set of all closed subsets of $X$. Turn $P$ and $Q$ into posets with the usual set-theoretic inclusion. Next, define $f^{*}:P\to Q$ by $f^{*}(U)=\overline{U}$, the closure of $U$, and $f_{*}:Q\to P$ by $f_{*}(V)=\mathrm{int}(V)$, the interior of $V$. Then $(f^{*},f_{*})$ is a Galois connection between $P$ and $Q$. Incidentally, those elements fixed by $f_{*}{f}^{*}$ are precisely the regular open sets of $X$, and those fixed by $f^{*}{f}_{*}$ are the regular closed sets.

Remark. The pair of functions in a Galois connection are order preserving as shown above. One may also define a Galois connection as a pair of maps $f^{*}:P\to Q$ and $f_{*}:Q\to P$ such that $f^{*}(p){\le }_{Q}q$ iff $f_{*}(q){\le }_{P}p$, so that the pair $f^{*},f_{*}$ are order reversing. In any case, the two definitions are equivalent in that one may go from one definition to another, (simply exchange $Q$ with $Q^{\partial }$, the dual (http://planetmath.org/DualPoset) of $Q$).