[[Category Theory MOC]]
# Commutative diagram

A **commutative diagram** is a directed graph with morphisms as edges and objects as vertices, such that any two paths sharing the same initial and final vertex are the same.[^br]
The diagram is thence said to **commute**.

[^br]: 2020, [[@bradleyTopologyCategoricalApproach2020|Topology: A categorical approach]], p. 4

## As a functor

A **diagram** of shape $\cat J$ in category $\cat C$ is a functor $\mathscr{D} : \cat J \to \cat C$,
where $\cat J$ is a usually [[Small category]] called the index category, #m/def/cat
and we typically use subscripts for objects.
Related concepts are [[Cones and cocones]], and then [[Limits and colimits]].


[^br]: 2020 [[@bradleyTopologyCategoricalApproach2020|Topology: A categorical approach]], §4.1, pp. 75–76

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