A galois connection

24 Dec 2015

Let $P,S$ be posets, with maps going back and forth $\varphi: P\to S$ and $\psi:S\to P$. Viewing the posets as categories and the maps as functors, an adjunction $\varphi \dashv \psi$ simply states that

$$\varphi (p) \leq s \Leftrightarrow p \leq \psi (s).$$

This follows from the fact that in this setup, a hom-set consists of at most one element, expressing an inequality. The naturality condition of the adjunction is trivial, a direct consequence of monotonicity of poset maps and transitivity of order.

An adjunction between posets goes by another name: a Galois connection (the fundamental theorem of Galois theory relates the lattices of certain subgroups and field extensions in this way). Let’s see a cute example from set theory.

Consider a set map $f:X\to Y$. It induces poset maps on the powersets: direct image $f_*: 2^X\to 2^Y$ and inverse image $f^*:2^Y\to 2^X$. Recall that for any subset $A\subseteq X$, we have the inclusion

$$A\subseteq f^*f_*A,$$

which I try to remember by the slogan «push-pull inflates» (decorated with suitable hand gestures). Dually, «pull-push deflates»:

$$f_*f^*B \subseteq B$$

for any $B\subseteq Y$.

These observations show that there is a Galois connection $\ f_* \dashv f^*$:

$f_* A \subseteq B \Leftrightarrow A \subseteq f^* B$.

One consequence of this is that direct image, as a left adjoint, preserves coproducts, AKA unions (but not necessarily intersection), while inverse image preserves intersections. This however does not explain why inverse image also preserves union… could it be that is is also left-adjoint?