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 Galois connection 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 $Q$).

Title	Galois connection
Canonical name	GaloisConnection
Date of creation	2013-03-22 15:08:15
Last modified on	2013-03-22 15:08:15
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	13
Author	CWoo (3771)
Entry type	Definition
Classification	msc 06A15
Synonym	Galois correspondence
Synonym	Galois connexion
Related topic	InteriorAxioms
Related topic	AdjointFunctor
Defines	upper adjoint
Defines	lower adjoint