## Encyclopedia > Commutative diagram

Article Content

# Commutative diagram

In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition.

For example, the first isomorphism theorem is a commutative triangle as follows:

Since f = h o φ, the left diagram is commutative; and since φ = k o f, so is the right diagram.

Similarly, the square above is commutative if y o w = z o x.

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.

All Wikipedia text is available under the terms of the GNU Free Documentation License

Search Encyclopedia
 Search over one million articles, find something about almost anything!

Featured Article
 Holtsville, New York ... (7.0 mi²). 18.0 km² (7.0 mi²) of it is land and none of the area is covered with water. Demographics As of the census of 2000, there are 17,006 people, ...