# Commutative diagram A commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result. It is said that commutative diagrams play the role in category theory that equations play in algebra. Commutative diagrams are a convenient way to visualize equalities of morphisms. We often use diagrams consisting of objects and morphisms of a category. Such a diagram in a category `𝒞` is said *to commute*, when, for each pair of objects `𝑎` and `𝑏`, any two paths leading from `𝑎` to `𝑏` yield, by composition, equal morphisms from `𝑎` to `𝑏`. Moreover, commutative diagrams are closed under composition of diagrams in that a diagram commutes if all of its subdiagrams commute. Saying a diagram commutes means that for all pairs of vertices, all paths from between them are equivalent, i.e. correspond to the same arrow of the category. --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://mandober.gitbook.io/math-debrief/450-category-theory/00-prereqs/commutative-diagram.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.