More topological reflections of arithmetic

For this part of the story I want a really-pretty diagram that even the mathjax still doesn't understand yet how to draw, so I've prepared a dead picture of it.

The doubled-arrows say what kinds of squares they cross --- what sort of equations. The whole thing is basically a fancy elaboration of the idea $\Phi + X = Z + Y$ and $\Psi Z = C \Phi$ and $A Z = X C$ and $A Y = X B$ and $B \Phi = Y \Psi$; and it says that if what it means by those are true, then (that's the dotted double arrow) also what it means by $\Psi + A = C + B$ is true. Note, however, that not all arithmetically true things are topologically true; in arithmetic we have: if $A B = 1$ and $C B = 1$, then $A = C$; but this isn't true in figures: one would need the right diagram to make it work.

The best known (if not best-known) generality described by the above diagram is known as “Mather's Second Cube Theorem”, and it's at the heart of a lovely little construction I mentioned earlier, which you get with $\Psi = *$ . There's essentially only one way for a space to “be $*$”, so these situations are rather special; on the other hand, as long as $\Phi$ all comes in one piece, there's exactly one way to fit any $A,B,C$ as in the diagram over any compatible $X, Y, Z$. What it meant for the earlier story is that one doesn't need to worry about $\Phi$ , because it might as well be $*$. Making that change doesn't even change the ... well, I can't quite say what it didn't change, because I haven't spelled it, here. But when you can simplify the problem without changing the solution, you're making progress!

There are lots more nifty diagrams involved, probably the prettiest being

--- the $\gg$ here is doing what the $\Downarrow$ was above, never mind. Anyways, I was going to write a fun little paper about it, but then I found this fine gem, which I think describes about the same argument, but better! This happens to me a lot, as you might imagine. One day...

Anyway, if anyone wants more, just ask me!