nLab
Galois connection

Definition

A Galois connection is a dual adjunction between posets.

More explicitly, given posets $A$ and $B$, a Galois connection between $A$ and $B$ is a pair of order-reversing maps $f:A\to B$ and $g:B\to A$ such that $a\le g(f(a))$ and $b\le f(g(b))$ for all $a\in A$, $b\in B$.

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

Examples

• Frequently Galois connections between collections of subsets 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 $y\in b$.” See orthogonality for one example.

• 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?.

Properties

Every Galois connection between full power sets,

$$(f:P(X)\to P(Y{)}^{\mathrm{op}})⊣(g:P(Y{)}^{\mathrm{op}}\to P(X))$$(f: P(X) \to P(Y)^{op}) \dashv (g: P(Y)^{op} \to P(X))

is of the form in the first example: there is some binary relation $r$ from $X$ to $Y$ such that

$$f(S)=\{y:{\forall }_{x\in X}x\in S\Rightarrow r(x,y)\},g(T)=\{x:{\forall }_{y\in Y}y\in T\Rightarrow r(x,y)\}$$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)\}

Indeed, define $r:X\times Y\to 2$ 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 $P(Y{)}^{\mathrm{op}}$, which are intersections in $P(Y)$. Since every $S$ in $P(X)$ is a union of singletons $\{x\}$, this gives

$$f(S)=\bigcap _{x\in S}f(\{x\})=\{y:{\forall }_{x\in S}r(x,y)\}$$f(S) = \bigcap_{x \in S} f(\{x\}) = \{y: \forall_{x \in S} r(x, y)\}

which is another way of writing the formula for $f$ given above. We observe that

$$T\subseteq f(S)=\{y:{\forall }_{x\in S}r(x,y)\}$$T \subseteq f(S) = \{y: \forall_{x \in S} r(x, y)\}

if and only if

$$S\times T\subseteq r$$S \times T \subseteq r

(now viewing $r$ extensionally in terms of subsets). This last symmetrical expression in $S$ and $T$ means

$$S\subseteq g(T):=\{x:{\forall }_{y\in T}r(x,y)\}$$S \subseteq g(T) := \{x: \forall_{y \in T} r(x, y)\}

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 $Y$.