commutative diagram (idea)

The basic commutative diagrams are:

A -> B       A -> B
|    |  and    \  |
v    v          v v
C -> D            C

To say that a diagram is commutative means the following. Suppose x is in A, and let A -> C -> ... -> B and A -> D -> ... -> B be two ways of chasing arrows, or paths, around the diagram (only going forward on the arrows). Then if the first path leads to y in C, the second path leads to the same y.

For example, let f be a map from A to B, g be a map from B to C, and h be a map from A to C. If the diagram

A -> B
  \  |
   v v
     C

commutes, then for any x in A, h(x)=g(f(x)).