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On TWF week 201, J. Baez explains the basics of Galois theory, and say at the end :

But here's the big secret: this has NOTHING TO DO WITH FIELDS! It works for ANY sort of mathematical gadget! If you've got a little gadget k sitting in a big gadget K, you get a "Galois group" Gal(K/k) consisting of symmetries of the big gadget that fix everything in the little one. But now here's the cool part, which is also very general. Any subgroup of Gal(K/k) gives a gadget containing k and contained in K: namely, the gadget consisting of all the elements of K that are fixed by everything in this subgroup. And conversely, any gadget containing k and contained in K gives a subgroup of Gal(K/k): namely, the group consisting of all the symmetries of K that fix every element of this gadget.

Apart from fields, what other "big gadgets" can be described in this way ? And what are the corresponding "little gadgets" ?

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There are several examples in Wikipedia. A famous example is the one between radical ideals and affine algebraic varieties. –  lhf Jan 13 '12 at 19:28
    
@lhf: actually what bothers me with these examples is that I don't see what are the corresponding "K" and "k". What would they be in the case of covering spaces, for example ? –  AlexPof Jan 13 '12 at 19:48
    
The title of your question suggests you are looking for examples of the general category-theoretic notion of a Galois connection; but from your comments below it seems that you want something more specific, with actual automorphism groups of sets with structure. Perhaps you could clarify? –  user16299 Jan 14 '12 at 23:55
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@Yemon Choi: as stated in another comment, I'm intrigued by the general statement of Baez which suggest that the functors always exist. So, yes, I'm actually more interested by the "automorphism groups of sets with structure" approach. Fields are a typical example of this approach but are there other examples ? (A lot of interesting galois connections were given below, but (I might be wrong), as said, I don't see the corresponding "K" and "k") –  AlexPof Jan 15 '12 at 22:57
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up vote 11 down vote accepted

There is a nice section on Galois connections in George Bergman's An Invitation to General Algebra and Universal Constructions (available from his website). If you want to learn more about Galois connections, I highly recommend reading the entire chapter (except perhaps section 5.4, which is a complete digression).

The general setting is:

Let $S$ and $T$ be sets, and let $R\subseteq S\times T$ be a relation from $S$ to $T$. For any $A\subseteq S$ and $B\subseteq T$, let $$\begin{align*} A^* &= \{t\in T\mid \forall a\in A (aRt)\}\subseteq T\\ B^* &= \{s\in S\mid \forall b\in B (sRb)\}\subseteq S. \end{align*}$$ This gives us two functions, one from $\mathcal{P}(S)$ to $\mathcal{P}(T)$, and one from $\mathcal{P}(T)$ to $\mathcal{P}(S)$. The operations are:

  1. Inclusion reversing: if $A\subseteq A'\subseteq S$ and $B\subseteq B'\subseteq T$, then $(A')*\subseteq A^*$ and $(B')^*\subseteq B*$.

  2. Increasing: $A\subseteq A^{**}$ for all $A\subseteq S$ and $B\subseteq B^{**}$ for all $B\subseteq T$.

  3. For all $A\subseteq S$, $A^{***}=A^*$; for all $B\subseteq T$, $B^{***}=B^*$.

In particular, the maps $A\mapsto A^{**}$ and $B\mapsto B^{**}$ give closure operators on $\mathcal{P}(S)$ and $\mathcal{P}(T)$. The closed subsets of $\mathcal{P}(S)$ are precisely the sets of the form $B^*$, the closed subsets of $\mathcal{P}(S)$ are precisely the sets of the form $B^*$, and the $*$ operation restricted to closed sets gives an anti-isomorphism (order-reversing) bijection between the complete lattice of ${}^{**}$-closed subsets of $S$ and of $T$.

Bergman notes:

A Galois connection between two sets $S$ and $T$ becomes particularly valuable when the ${}^{**}$-closed subsets have characterizations of independent interest.

Here are the examples he gives:

  1. The "classical example". Let $S$ be the underlying set of a field $F$, $T$ the underlying set of a finite group $G$ of automorphisms of $F$. For $a\in F$ and $g\in G$, let $aRg$ mean "$g$ fixes $a$" (i.e., $g(a)=a$). The Fundamental Theorem of Galois Theory says that the closed subsets of $F$ are precisely the subfields of $F$ that contain the set $G^*$, and that the closed subsets of $G$ are precisely all subgroups of $G$.

  2. Let $S$ be a vector space over a field $K$, and $T$ the dual space $\mathrm{Hom}_K(S,K)$. Let $xRf$ mean "$f(x)=0$". The closed subsets of $S$ are precisely the vector subspaces, the closed subsets of $T$ are precisely the vector subspaces that are closed in a certain topology (all subspaces when the dimension is finite, but when the dimension is infinite you get interesting stuff).

  3. Let $S=\mathbb{C}^n$ (complex affine $n$-space), and $T=\mathbb{Q}[x_0,\ldots,x_{n-1}]$, the polynomial ring in $n$ indeterminates with rational coefficients. Let $(a_0,\ldots,a_{n-1})Rf$ mean $f(a_0,\ldots,a_{n-1})=0$. This is the starting point of classical algebraic geometry: the closed subsets of $\mathbb{C}^n$ are the solution sets of polynomial equations, and the Nullstellensatz characterizes the closed subsets of $T$ as the radical ideals.

  4. Let $S$ be a finite dimensional real vector space, $T$ the set of pairs $(f,a)$ where $f$ is a linear functional on $S$ and $a\in\mathbb{R}$. Let $xR(f,a)$ mean $f(x)\leq a$. The closed subsets of $S$ are the convex sets.

  5. Let $S$ be a finite dimensional vector space, $T$ the set of linear functionals on $S$. Let $xRf$ mean $f(x)\leq 1$. The closed subsets on each side are the convex subsets that contain the zero vector.

  6. Let $A$ be an abelian group (or a module over a commutative ring) and $S=T=\mathrm{End}(A)$ the ring of endomorphisms. Let $sRt$ stand for $st=ts$. Given a subring $X$ of $S$, $X^*$ is the commutatnt of $X$, an important subring studied by ring theorists.

  7. Let $S$ be a set of mathematical objects, $T$ a set of propositions about objects of this sort, and $sRt$ stand for "object $s$ satisfies proposition $t$" (for logicians, $s\models t$). The closed subsets of $S$ are the axiomatic classes, the closed subsets of $T$ are the theories.


Added. It seems to me that Baez is staking a middle ground between the full generality of Galois connections and the special case of Galois Theory of fields, by considering only the situation in which $T$ is a set, $S$ is a subgroup of the group of bijections $T\to T$, and the relation is $fRt$ if and only if $f(t)=t$. In that case, you can always let $k=S^*$, and you do indeed get a correspondence between the closed subsets of $T$ and the closed subgroups of $S$ (though not every subgroup need be closed; this occurs for instance in the Galois group of an infinite field extension, where only subgroups that are closed in the profinite topology correspond to subfield of the extensions).

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If you take the lattice of ideals of a ring $R$ and the lattice of ideals of $R/\mathfrak{a}$ for some ideal $\mathfrak{a}$ then the maps $\mathfrak{b}\mapsto\pi(\mathfrak{b})$ and it's inverse form a monotone Galois connection between the ideals of $R$ containing $\mathfrak{a}$ and the ideals of $R/\mathfrak{a}$.

I'm sure you can make a similar guess for the case of subgroups of groups, submodules of modules, etc.

More generally, a Galois connection between two posets $I$ and $J$ is nothing more than an adjoint pair of functors between their corresponding categories. This allows you to create a whole host of examples.

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I agree with the general definition of a Galois connection, but it sounds to me like an a posteriori observation on its structure. But a priori, one still has to prove the correspondance between intermediate extension of fields and subgroups of the Galois group, for example. What intrigues me is the statement of J. Baez which suggests that these functors always exist when you look at the internal symmetries of your "K"... –  AlexPof Jan 13 '12 at 19:54
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@AlexPof I don't think that Baez was being rigorous. In my experience Baez's TWF series is more expository, and often foregoes formality for understandability. I agree that in general I don't see how to construct what he suggests is constructible. –  Alex Youcis Jan 13 '12 at 19:57
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Covering spaces and fundamental groups.

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