Commutative diagram

In mathematics, especially homological algebra and category theory, a commutative diagram is a diagram of objects and morphisms so that, when picking two objects, one can follow any path through the diagram and obtain the same result. For example, the first isomorphism theorem is a commutative triangle as follows:

Since f = φh, the left diagram is commutative; and since φ = fk, so is the right diagram.

Similarly, the square above is commutative if yw = zx.

Commutativity for general diagrams can be recursively defined by stating that all n-sided polygonal subdiagrams must be commutative n-gons for any n.