TITLE: Examples of Galois connections?
QUESTION [18 upvotes]: 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" ?

REPLY [5 votes]: Let $(\Omega,\Sigma, \mathbb P)$ be a probability space and let $X : \Omega \to \mathbb R$ be a random variable. Consider the cumulative distribution function: $$F_X : [-\infty\,.. \infty] \to [0\,.. 1], x\mapsto \mathbb P (X\leq x)$$
This is a monotone map and it always has left-adjoint $Q_X$ called the "quantile function" given by:
$$Q_X : [0\,.. 1] \to [-\infty\,.. \infty] , p\mapsto \inf\{x\in \mathbb R : p< F(x)\}$$
For example the median of $X$ would be $Q_X(\frac{1}{2})$. 
Since adjoints are unique (up to natural isomorphism, but here we have posets) the quantile function determines the distribution of $X$ much like the CDF, PDF, PMF or characteristic function.
The quantile function is also known as "inverse distribution function", "generalized inverse function" and so on and so forth. As far as I can guess it is simply not well-known that this is simply an instance of a Galois connection.