Commutative diagram

A diagram of objects and maps in which every pair of directed paths with the same endpoints composes to the same map. It expresses equality of composites visually.

Category: A structure consisting of objects, morphisms between them, and an associative composition operation with identity morphisms. A commutative diagram is drawn inside a category, whose composition determines what its paths mean.
Universal property: A characterization of an object by the unique maps it admits to or from other objects. Universal properties are commonly expressed as commutative diagrams with a uniqueness condition.
Commutative square: A four-sided diagram whose two directed routes between opposite corners compose to the same map. It is the simplest common shape in which distinct paths must agree.
Exact diagram: A diagram in which specified consecutive maps have image equal to the next map's kernel. Exactness imposes image-kernel conditions beyond the path equalities of commutativity.
Natural transformation: A family of morphisms between two functors that is compatible with every morphism in their source category. Its compatibility condition is expressed by a commuting square for each source morphism.
Morphism: A structure-preserving map between objects in a mathematical setting such as a category. Each arrow represents a morphism whose composition can be compared with other paths.
Exact sequence: A sequence of maps in which each map's image equals the next map's kernel. Diagrams organize exact sequences and show how their maps interact.
Commutative triangle: A triangular diagram in which a direct map equals the composite along the other two sides. It expresses path equality with one direct arrow and one two-step route.
Anticommutative diagram: A diagram in which specified composites agree up to a minus sign rather than being equal. It replaces selected commutativity equations with sign-adjusted ones.
Yoneda lemma: A theorem relating maps into an object to natural transformations from its represented functor. Its proof and applications are commonly organized through naturality and commutative diagrams.

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