nLab
Galois connection

Definition

In order theory the term Galois connection 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”.

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 $Y$.