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 of 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 function
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 a projective module iff any surjective module
homomorphism 
and any module homomorphism 
can be completed to a commutative diagram.
Similarly, one can characterize the dual notion of an injective module: a module 
is an injective module iff any injective module
homomorphism 
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 the ring 
in 
.
A chain homomorphism can be visualized as a sequence of commutative squares. The displayed boundary
operators point toward increasing indices. With the homological indexing convention
,
their arrows instead point toward decreasing indices.
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."
