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)).