Commutative diagram

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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 endpoints lead to the same result by composition. Commutative diagrams play the role in category theory that equations play in algebra.

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.

Contents

Examples

In the following diagram expressing the first isomorphism theorem, commutativity means that f = \tilde{f} \circ \pi:

First isomorphism theorem (plain).svg

Below is a generic commutative square, in which h \circ f = k \circ g

Commutative square.svg

Symbols

In algebra texts, the type of morphism can be denoted with different arrow usages: monomorphisms with a \hookrightarrow, epimorphisms with a \twoheadrightarrow, and isomorphisms with a \overset{\sim}{\rightarrow}. The dashed arrow typically represents the claim that the indicated morphism exists whenever the rest of the diagram holds. This is common enough that texts often do not explain the meanings of the different types of arrow.

Verifying commutativity

Commutativity makes sense for a polygon of any finite number of sides (including just 1 or 2), and a diagram is commutative if every polygonal subdiagram is commutative.

Diagram chasing

Diagram chasing is a method of mathematical proof used especially in homological algebra. Given a commutative diagram, a proof by diagram chasing involves the formal use of the properties of the diagram, such as injective or surjective maps, or exact sequences. A syllogism is constructed, for which the graphical display of the diagram is just a visual aid. It follows that one ends up "chasing" elements around the diagram, until the desired element or result is constructed or verified.

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