Commutative diagram
From Wikipedia, the free encyclopedia
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. Commutative diagrams play the role in category theory that equations play in algebra.
Contents |
[edit] Examples
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.
[edit] 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.
[edit] 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 formally using 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. One ends up "chasing" elements around the diagram, until the desired element or result is constructed or verified.
Examples of proofs by diagram chasing include those typically given for the five lemma, the snake lemma, the zig-zag lemma, and the nine lemma.
