Commutative diagram

In mathematics, and especially in category theory, a commutative diagram is a diagram of objects (also known as vertices) and morphisms (also known as arrows or edges) such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition. Commutative diagrams play the role in category theory that equations play in algebra (see Barr-Wells, Section 1.7).

Note that a diagram may not be commutative, i.e., the composition of different paths in the diagram may not give the same result. For clarification, phrases like "this commutative diagram" or "the diagram commutes" may be used.

In the bottom-left diagram, which expresses the first isomorphism theorem, commutativity means that ${\displaystyle f={\tilde {f}}\circ \pi }$ while in the bottom-right diagram, commutativity of the square means ${\displaystyle h\circ f=k\circ g}$:

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