Category theory: Enough that polygonal diagrams commute I've read somewhere that for a categorical diagram to commute, it is enough that all its polygonal subdiagrams commute.
I want a reference and a detailed proof of this.
Please also give a formal definition of polygonal subdiagrams.
 A: I myself have been recently looking for a citation and a proof, but here (I think) is the idea behind it.
First, to clear up the question, I will assume you are referencing the brief tidbit mentioned in the commutative diagram page on Wikipedia under "verifying commutativity" (https://en.m.wikipedia.org/wiki/Commutative_diagram). This means you are talking about a result concerning diagrams and commutativity generally, not just ones representing categories (at least conceptually). In a hand-wavey way, we assume each of the planar faces of subdiagrams commute and want to show the whole diagram commutes. 
So assume all the polygonal subdiagrams in a diagram commute. Formally I think you should proceed by induction on the number of vertices. The base case is trivial. For the inductive step you pick any two vertices and show any two paths to those two vertices are the same. You can do this since any proper subpath is the same route by inductive hypothesis, extending a path and showing it's the same is fine, and paths which are equal to the same path are equal to one another. 
Try working through a diagram with only four vertices and some triangles; the general procedure should become clear. 
