Maybe you've heard about something called a Chu space around here. But what the heck is a Chu space? And whatever it is, does it really belong with all the rich mathematical structures we know and love?

Say you have some stuff. What can you do with it?

Maybe it's made of little pieces, and you can do a different thing with each little piece.

But maybe the pieces are structured in a certain way, and you aren't allowed to do anything that would break this structure.

A Chu space is a versatile way of formalizing anything like that!

To represent something in a Chu space, we'll put the names of our pieces on the left. How about the rules? For a Chu space, the rules are about allowed ways to color our pieces. To represent these rules, we can simply make columns showing all the allowed ways we can color our pieces (just get rid of any columns that break the rules). Here's what a basic 3-element set (pictured on the left) looks like as a Chu space:

It doesn't have any sort of structure, so we show that by allowing all the possible colorings (with two colors). Chu spaces that don't have any rules (i.e. all colorings are allowed) are equivalent to sets.

What about the one with the arrows from above? How can we make an arrow into a coloring rule? One way we could do it is by stipulating that if there's an arrow $x \to y$ , we'll make a rule that if $x$ is colored black, then $y$ has to be colored black too, where $x$ and $y$ can stand in for any of the pieces. Here's what that Chu space looks like:

Spend a minute looking at the picture until you're convinced that our coloring rule is obeyed for every arrow on the left side of the picture. Any Chu space that has this kind of arrow rule has the structure of a poset.

There and back again

If we have two Chu spaces, say $A$ and $B$ , what sort of maps should we be able to make between them?

We'd like to be able to map the pieces of $A$ to the pieces of $B$ . So this part of our map will just be a normal function between sets:

$f : A_{pieces} \to B_{pieces}$

But we also want our maps to respect the rules of $A$ as it maps to $B$ . How can we check this?

Let's think through how we would be able to tell if a potential map did break the rules. The map will take pieces of $A$ to pieces of $B$ , so let's look at the pieces of $B$ that got mapped onto, i.e. $f (A_{pieces})$ . It will help if we have a concrete example in mind, so let's consider a potential map that we think should break the rule...

Will Ferell

Will Ferell

Will Ferell

Will Ferell