Lecture 4 - Galois Connections

Okay, now let's get to the cool part: Galois connections. Before he died in a duel, the young Évariste Galois proved that you couldn't solve the quintic equation with radicals: that is, there's nothing like the quadratic formula for equations like

[ ax^5 + bx^4 + cx^3 + dx^2 + ex + f = 0. ]

He used a trick for converting one view of a problem into another, and then converting the other view back into the original one. By now, we've extracted the essence of this trick and dubbed it a "Galois connection". It's far more general than Galois dreamed.

Remember, a preorder is a set \(A\) with a relation \(\le_A\) that's reflexive and transitive. When we're in the mood for being careful, we write a preorder as a pair \( (A,\le_A)\). When we're feeling lazy we'll just call it something like \(A\), and just write \(\le\) for the relation.

Definition. Given preorders \((A,\le_A)\) and \((B,\le_B)\), a monotone function from \(A\) to \(B\) is a function \(f : A \to B\) such that

[ x \le_A y \textrm{ implies } f(x) \le_B f(y) ]

for all elements \(x,y \in A\),

Puzzle 10. There are many examples of monotone maps between preorders. List a few interesting ones!

Definition. Given preorders \((A,\le_A)\) and \((B,\le_B)\), a Galois connection is a monotone function \(f : A \to B\) together with a monotone function \(g: B \to A\) such that

[ f(a) \le_B b \textrm{ if and only if } a \le_A g(b) ]

for all \(a \in A, b \in B\). In this situation we call \(f\) the left adjoint and \(g\) the right adjoint.

So, the right adjoint of \(f\) is a way of going back from \(B\) to \(A\) that's related to \(f\) in some way.

Puzzle 11. Show that if the monotone function \(f: A \to B\) has an inverse \(g : B \to A \) that is also a monotone function, then \(g\) is both a right adjoint and a left adjoint of \(f\).

So, adjoints are some sort of generalization of inverses. But as you'll eventually see, they're much more exciting!

I will spend quite a few lectures describing really interesting examples, and you'll start seeing what Galois connections are good for. It shouldn't be obvious yet, unless you already happen to know or you're some sort of superhuman genius. I just want to get the definition on the table right away.

Here's one easy example to get you started. Let \(\mathbb{N}\) be the set of natural numbers with its usual notion of \(\le\). There's a function \(f : \mathbb{N} \to \mathbb{N}\) with \(f(x) = 2x \). This function doesn't have an inverse. But:

Puzzle 12. Find a right adjoint for \(f\): that is, a function \(g : \mathbb{N} \to \mathbb{N}\) with

[ f(m) \le n \textrm{ if and only if } m \le g(n) ]

for all \(m,n \in \mathbb{N}\). How many right adjoints can you find?

Puzzle 13. Find a left adjoint for \(f\): that is, a function \(g : \mathbb{N} \to \mathbb{N}\) with

[ g(m) \le n \textrm{ if and only if } m \le f(n) ]

for all \(m,n \in \mathbb{N}\). How many left adjoints can you find?

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