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\section*{Galois connection}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory}

[[!include category theory - contents]]

\hypertarget{galois_connections}{}\section*{{Galois connections}}\label{galois_connections}

\noindent\hyperlink{Idea}{Idea}\dotfill \pageref*{Idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{GaloisTheory}{Galois theory}\dotfill \pageref*{GaloisTheory} \linebreak
\noindent\hyperlink{InducedFromARelation}{Galois connections induced from a relation}\dotfill \pageref*{InducedFromARelation} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{BetweenPowerSets}{Galois connections between power sets}\dotfill \pageref*{BetweenPowerSets} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{Idea}{}\subsection*{{Idea}}\label{Idea}

In [[order theory]] the term \emph{Galois connection} (due to \hyperlink{Ore44}{Ore 44}, who spelled it ``connexion'') can mean both: ``[[adjunction]] between [[posets]]'' and ``[[dual adjunction]] between [[posets]]''; the former notion is sometimes called ``monotone Galois connection'' and the latter ``antitone Galois connection''. In this article the term ``Galois connection'' shall mean ``[[dual adjunction]] between [[posets]]''.

The term \emph{Galois correspondence} is also in use. For some authors it is synonymous to ``Galois connection'', others reserve it for its restriction to its [[fixed point of an adjunction|fixed points]], where it becomes an [[adjoint equivalence]].

The example that gives the concept its name is the relation between [[subgroups]] and [[subfields]] in [[Galois theory]] (see \hyperlink{GaloisTheory}{below}), but adjunctions between posets, hence Galois connections, appear also in many other and entirely different contexts, see further \hyperlink{Examples}{below}.

\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

Given [[posets]] $A$ and $B$, a \textbf{Galois connection} between $A$ and $B$ is a pair of order-reversing [[functions]] $f \colon A\to B$ and $g \colon B\to A$ such that $a\le g(f(a))$ and $b\le f(g(b))$ for all $a\in A$, $b\in B$.

A \textbf{Galois correspondence} is a Galois connection which is an [[adjoint equivalence]] (so $a = g(f(a))$ and $b = f(g(b))$ for all $a \in A$, $b \in B$).

\begin{prop}
\label{}\hypertarget{}{}
Any Galois connection $f: A \to B$, $g: B \to A$ induces a Galois correspondence between $f(A)$ and $g(B)$, given by the composites $g(B) \hookrightarrow A \stackrel{f}{\to} f(A)$ and $f(A) \hookrightarrow B \stackrel{g}{\to} g(B)$.

\end{prop}
\begin{proof}
For any $a \in A$ of the form $a = g(b)$, we have $a \leq (g \circ f)(a)$ and also $(g \circ f)(a) = g(f(g(b))) \leq g(b) = a$ where the inequality follows from $b \leq f(g(b))$ and antitonicity of $g$. Hence $(g \circ f)(a) = a$ for all $a \in g(B)$. Similarly $(f \circ g)(b) = b$ for all $b \in f(A)$.

\end{proof}
\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

\hypertarget{GaloisTheory}{}\subsubsection*{{Galois theory}}\label{GaloisTheory}

The [[Galois theory]] normally taught in graduate-level algebra courses (and based on the work of [[Évariste Galois]]) involves a Galois connection between the intermediate [[fields]] of a [[Galois extension]] and the subgroups of the corresponding [[Galois group]].

\hypertarget{InducedFromARelation}{}\subsubsection*{{Galois connections induced from a relation}}\label{InducedFromARelation}

Frequently Galois connections between collections of [[subsets]] ([[power sets]]) arise where $f(a)$ is ``the set of all $y$ standing in some relation to every $x\in a$'' and dually $g(b)$ is ``the set of all $x$ standing in some relation to every $y\in b$.''

Examples of this class of Galois connections include the following

\begin{itemize}%
\item \textbf{(Zariski topology)} The [[closed subsets]] in the [[Zariski topology]] on [[affine space]] $k^n$ or on the set of [[maximal ideals]] of a [[polynomial ring]], which may be understood as the [[fixed point of an adjunction|fixed points]] of a Galois connection between [[polynomials]] and [[affine space]]/[[maximal ideal]]. This is discussed at \emph{\href{Zariski+topology#InTermsOfGaloisConnections}{Zariski topology -- In terms of Galois connections}}.


\item \textbf{(orthogonality classes)} Given a [[category]] $\mathcal{C}$, then on the [[poset]] of sub-[[classes]] of [[morphisms]] the operations of forming left and right classes with [[orthogonality]] [[lifting property]] constitute a Galois connection.



\end{itemize}
In fact all Galois connections between [[power sets]] arise this way, see \hyperlink{BetweenPowerSets}{below}.

We now spell out in detail the Galois connections induced from a relation:

\begin{defn}
\label{GaloisConnectionFromRelation}\hypertarget{GaloisConnectionFromRelation}{}
\textbf{(Galois connection induced from a [[relation]])}

Consider two [[sets]] $X,Y \in Set$ and a [[relation]]

\begin{displaymath}
E \hookrightarrow X \times Y
  \,.
\end{displaymath}
Define two [[functions]] between their [[power sets]] $P(X), P(Y)$, as follows. (In the following we write $E(x, y)$ to abbreviate the formula $(x, y) \in E$.)

\begin{enumerate}%
\item Define

\begin{displaymath}
V_E
    \;\colon\;
  P(X)
    \longrightarrow
  P(Y)
\end{displaymath}
by

\begin{displaymath}
V_E(S)
    \coloneqq
  \left\{
    y \in Y \vert  \underset{x \in  X}{\forall} \left( \left(x \in S\right) \Rightarrow E(x, y)
 \right) \right\}
\end{displaymath}

\item Define

\begin{displaymath}
I_E
    \;\colon\;
  P(Y)
    \longrightarrow
  P(X)
\end{displaymath}
by

\begin{displaymath}
I_E(T) \coloneqq \left\{x \in X \vert \underset{y \in Y}{\forall} \left( \left(y \in T \right) \Rightarrow E(x, y)  \right)\right\}
\end{displaymath}


\end{enumerate}
\end{defn}
\begin{prop}
\label{GaloisConnectionAsAdjunction}\hypertarget{GaloisConnectionAsAdjunction}{}
The construction in def. \ref{GaloisConnectionFromRelation} has the following properties:

\begin{enumerate}%
\item $V_E$ and $I_E$ are [[contravariant functor|contravariant]] order-preserving in that

\begin{enumerate}%
\item if $S \subset S'$, then $V_E(S') \subset V_E(S)$;


\item if $T \subset T'$, then $I_E(T') \subset I_E(T)$



\end{enumerate}

\item The \emph{[[adjunction]] law} holds: $\left(
     T \subset V_E(S)
  \right)
    \,\Rightarrow\,
  \left(
    S \subset I_E(T)
  \right)$

which we denote by writing

\begin{displaymath}
P(X)
    \underoverset{\underset{V_E}{\longrightarrow}}{\overset{I_E}{\longleftarrow}}{\bot}
  P(Y)^{op}
\end{displaymath}

\item both $V_E$ as well as $I_E$ take [[unions]] to [[intersections]].



\end{enumerate}
\end{prop}
\begin{proof}
Regarding the first point: the larger $S$ is, the more conditions that are placed on $y$ in order to belong to $V_E(S)$, and so the smaller $V_E(S)$ will be.

Regarding the second point: This is because both these conditions are equivalent to the condition $S \times T \subset E$.

Regarding the third point: Observe that in a poset such as $P(Y)$, we have that $A = B$ iff for all $C$, $C \leq A$ iff $C \leq B$ (this is the [[Yoneda lemma]] applied to posets). It follows that

\begin{displaymath}
\itexarray{
    T \subset V_E(\bigcup_{i \in I} S_i) & \text{iff} & \bigcup_{i: I} S_i \subset I_E(T) \\
    & \text{iff} & \forall_{i: I} S_i \subset I_E(T) \\
    & \text{iff} & \forall_{i: I} T \subset V_E(S_i) \\
    & \text{iff} & T \subset \bigcap_{i: I} V_E(S_i)
}
\end{displaymath}
and we conclude $V_E(\bigcup_{i: I} S_i) = \bigcap_{i: I} V_E(S_i)$ by the [[Yoneda lemma]].

\end{proof}
\begin{prop}
\label{GaloisClosureOperator}\hypertarget{GaloisClosureOperator}{}
\textbf{([[closure operators]] from Galois connection)}

Given a Galois connection as in def. \ref{GaloisConnectionFromRelation}, consider the [[composition|composites]]

\begin{displaymath}
I_E \circ V_E \;\colon\; P(X) \longrightarrow P(X)
\end{displaymath}
and

\begin{displaymath}
V_E \circ I_E \;\colon\; P(Y) \longrightarrow P(Y)
  \,.
\end{displaymath}
These satisfy:

\begin{enumerate}%
\item For all $S \in P(X)$ then $S \subset I_E \circ V_E(S)$.


\item For all $S \in P(X)$ then $V_E \circ I_E \circ V_E (S)  =  V_E(S)$.


\item $I_E \circ V_E$ is [[idempotent]] and [[covariant functor|covariant]].



\end{enumerate}
and

\begin{enumerate}%
\item For all $T \in P(Y)$ then $T \subset V_E \circ I_E(T)$.


\item For all $T \in P(Y)$ then $I_E \circ V_E \circ I_E (T) = I_E(T)$.


\item $V_E \circ I_E$ is [[idempotent]] and [[covariant functor|covariant]].



\end{enumerate}
This is summarized by saying that $I_E \circ V_E$ and $V_E \circ I_E$ are \emph{[[closure operators]]} ([[idempotent monads]]).

\end{prop}
\begin{proof}
The first statement is immediate from the adjunction law (prop. \ref{GaloisConnectionAsAdjunction}).

Regarding the second statement: This holds because applied to sets $S$ of the form $I_E(T)$, we see $I_E(T) \subset I_E \circ V_E \circ I_E(T)$. But applying the contravariant map $I_E$ to the inclusion $T \subset V_E \circ I_E(T)$, we also have $I_E \circ V_E \circ I_E(T) \subset I_E(T)$.

This directly implies that the function $I_E \circ V_E$. is idempotent, hence the third statement.

The argument for $V_E \circ I_E$ is directly analogous.

\end{proof}
In view of prop. \ref{GaloisClosureOperator} we say that:

\begin{defn}
\label{GaloisClosedElements}\hypertarget{GaloisClosedElements}{}
\textbf{(closed elements)}

Given a Galois connection induced from a relation as in def. \ref{GaloisConnectionFromRelation}, then

\begin{enumerate}%
\item $S \in P(X)$ is called \emph{closed} if $I_E \circ V_E(S) = S$;


\item the \emph{closure} of $S \in P(X)$ is $Cl(S) \coloneqq I_E \circ V_E(S)$



\end{enumerate}
and similarly

\begin{enumerate}%
\item $T \in P(Y)$ is called \emph{closed} if $V_E \circ I_E(T) = T$;


\item the \emph{closure} of $T \in P(Y)$ is $Cl(T) \coloneqq V_E \circ I_E(T)$.



\end{enumerate}
\end{defn}
It follows from the properties of [[closure operators]], hence form prop. \ref{GaloisClosureOperator}:

\begin{prop}
\label{GaloisFixedPoints}\hypertarget{GaloisFixedPoints}{}
\textbf{([[fixed point of an adjunction|fixed points]] of a [[Galois connection]])}

Given a Galois connection induced from a relation as in def. \ref{GaloisConnectionFromRelation}, then

\begin{enumerate}%
\item the closed elements of $P(X)$ are precisely those in the [[image]] $im(I_E)$ of $I_E$;


\item the closed elements of $P(Y)$ are precisely those in the [[image]] $im(V_E)$ of $V_E$.



\end{enumerate}
We says these are the \emph{[[fixed point of an adjunction|fixed points]]} of the Galois connection. Therefore the restriction of the Galois connection

\begin{displaymath}
P(X)
    \underoverset{\underset{V_E}{\longrightarrow}}{\overset{I_E}{\longleftarrow}}{\bot}
  P(Y)^{op}
\end{displaymath}
to these fixed points yields an [[equivalence of categories|equivalence]]

\begin{displaymath}
im(I_E)
    \underoverset{\underset{V_E}{\longrightarrow}}{\overset{I_E}{\longleftarrow}}{\simeq}
  im(V_E)^{op}
\end{displaymath}
now called a \emph{[[Galois correspondence]]}.

\end{prop}
\begin{prop}
\label{}\hypertarget{}{}
Given a Galois connection induced from a relation as in def. \ref{GaloisConnectionFromRelation}, then the sets of closed elements according to def. \ref{GaloisClosedElements} are closed under forming [[intersections]].

\end{prop}
\begin{proof}
If $\{T_i \in P(Y)\}_{i: I}$ is a collection of elements closed under the operator $K = V_E \circ I_E$, then by the first item in prop. \ref{GaloisClosureOperator} it is automatic that $\bigcap_{i: I} T_i \subset K(\bigcap_{i: I} T_i)$, so it suffices to prove the reverse inclusion. But since $\bigcap_{i: I} T_i \subset T_i$ for all $i$ and $K$ is covariant and $T_i$ is closed, we have $K(\bigcap_{i: I} T_i) \subset K(T_i) \subset T_i$ for all $i$, and $K(\bigcap_{i: I} T_i) \subset \bigcap_{i: I} T_i$ follows.

\end{proof}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{BetweenPowerSets}{}\subsubsection*{{Galois connections between power sets}}\label{BetweenPowerSets}

\emph{Every} Galois connection between full [[power sets]],

\begin{displaymath}
(f: P(X) \to P(Y)^{op}) \dashv (g: P(Y)^{op} \to P(X))
\end{displaymath}
is of the form in def. \ref{GaloisConnectionFromRelation} \hyperlink{InducedFromARelation}{above}: there is some [[binary relation]] $r$ from $X$ to $Y$ such that

\begin{displaymath}
f(S) = \{y: \forall_{x \in X} x \in S \Rightarrow r(x, y)\}, \qquad g(T) = \{x: \forall_{y \in Y} y \in T \Rightarrow r(x, y)\}
\end{displaymath}
Indeed, define $r: X \times Y \to \mathbf{2}$ by stipulating that $r(x, y)$ is true if and only if $y \in f(\{x\})$. Because $f$ is a left adjoint, it takes colimits in $P(X)$ (in this case, unions) to colimits in $P(Y)^{op}$, which are intersections in $P(Y)$. Since every $S$ in $P(X)$ is a union of singletons $\{x\}$, this gives

\begin{displaymath}
f(S) = \bigcap_{x \in S} f(\{x\}) = \{y: \forall_{x \in S} r(x, y)\}
\end{displaymath}
which is another way of writing the formula for $f$ given above. We observe that

\begin{displaymath}
T \subseteq f(S) = \{y: \forall_{x \in S} r(x, y)\}
\end{displaymath}
if and only if

\begin{displaymath}
S \times T \subseteq r
\end{displaymath}
(now viewing $r$ extensionally in terms of subsets). This last symmetrical expression in $S$ and $T$ means

\begin{displaymath}
S \subseteq g(T) := \{x: \forall_{y \in T} r(x, y)\}
\end{displaymath}
which means we have a Galois connection between $f$ and $g$ under this definition; since $g$ is uniquely determined by the presence of a Galois connection with $f$, we conclude that all Galois connections between power sets arise in this way, via a relation $r$ between $X$ and $Y$.

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[modality]]
\item [[adjoint modality]]
\item [[adjunction]]
\item Every Galois connection is an [[idempotent adjunction]].
\item [[nucleus of a profunctor]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The concent is due to

\begin{itemize}%
\item \O{}ystein Ore, \emph{Galois connexions} , Trans. AMS \textbf{55} (1944) pp.493-513. (\href{http://www.ams.org/journals/tran/1944-055-00/S0002-9947-1944-0010555-7/S0002-9947-1944-0010555-7.pdf}{pdf})

\end{itemize}
Introduction is in

\begin{itemize}%
\item M. Ern\'e{}, E. Klossoswki, A. Melton, G. E. Strecker, \emph{A primer in Galois connections} , Annals New York Academy of Sciences \textbf{704} (1993) pp.103-125. (\href{http://www.math.ksu.edu/~strecker/primer.ps}{draft})

\end{itemize}
See also

\begin{itemize}%
\item Wojciech Dzik, Jouni J\"a{}rvinen, Michiro Kondo, \emph{Characterising intermediate tense logics in terms of Galois connections} (\href{http://arxiv.org/abs/1401.7646}{arXiv:1401.7646}).

\end{itemize}
[[!redirects Galois connection]] [[!redirects Galois connections]] [[!redirects Galois correspondence]] [[!redirects Galois correspondences]]



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