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.¹ The diagram is thence said to commute. For example, to say

commutes is to say that $𝑔 𝑓 = ℎ$ .

As a functor

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

#state/tidy | #lang/en | #SemBr | #lang/en

2020, Topology: A categorical approach, p. 4 ↩

Commutative diagram

As a functor

Footnotes