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Concept
Commutative diagram
Summary
In mathematics, and especially in , a commutative diagram is a 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.
A commutative diagram often consists of three parts:
(also known as vertices)
morphisms (also known as arrows or edges)
paths or composites
In algebra texts, the type of morphism can be denoted with different arrow usages:
A monomorphism may be labeled with a or a .
An epimorphism may be labeled with a .
An isomorphism may be labeled with a .
The dashed arrow typically represents the claim that the indicated morphism exists (whenever the rest of the diagram holds); the arrow may be optionally labeled as .
If the morphism is in addition unique, then the dashed arrow may be labeled or .
The meanings of different arrows are not entirely standardized: the arrows used for monomorphisms, epimorphisms, and isomorphisms are also used for injections, surjections, and bijections, as well as the cofibrations, fibrations, and weak equivalences in a .
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.
Note that a diagram may be non-commutative, i.e., the composition of different paths in the diagram may not give the same result.
In the left diagram, which expresses the first isomorphism theorem, commutativity of the triangle means that . In the right diagram, commutativity of the square means .
In order for the diagram below to commute, three equalities must be satisfied:
Here, since the first equality follows from the last two, it suffices to show that (2) and (3) are true in order for the diagram to commute. However, since equality (3) generally does not follow from the other two, it is generally not enough to have only equalities (1) and (2) if one were to show that the diagram commutes.
Official source
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