Commutative Diagram

A commutative diagram is a collection of maps  in which all map compositions starting from the same set  and ending with the same set  give the same result. In symbols this means that, whenever one can form two sequences

(1)

and

(2)

the following equality holds:

(3)

Commutative diagrams are usually composed by commutative triangles and commutative squares.

Commutative triangles and squares can also be combined to form plane figures or space arrangements.

A commutative diagram can also contain multiple arrows that indicate different maps between the same two sets.

A looped arrow indicates a map from a set to itself.

The above commutative diagram expresses the fact that  is the inverse map to , since it is a pictorial translation of the map equalities  and .

This can also be represented using two separate diagrams.

Many other mathematical concepts and properties, especially in algebraic topology, homological algebra, and category theory, can be formulated in terms of commutative diagrams.

For example, a module  is projective iff any surjective module homomorphism  and any module homomorphism  can be completed to a commutative diagram.

Similarly, one can characterize the dual notion of injective module: a module  is injective iff any injective module homomorphism and  and any module homomorphism  can be completed to a commutative triangle.

According to Baer's criterion, it is sufficient to require this condition for the inclusion maps  of the ideals of  in .

Another example of a notion based on diagrams is the chain homomorphism, which can be visualized as a sequence of commutative squares.

The advantage of drawing commutative diagrams is the possibility to seize any given map configuration at a glance. The picture also facilitates the task of composing maps, which is like following directed paths from set to set. Many homological theorems are proven by studying commutative diagrams: this method is usually referred to as "diagram chasing."

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