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!