- For help on drawing commutative diagrams on Wikipedia, see meta:Help:Displaying a formula#Commutative_diagrams.
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 directed path through the diagram and obtain the same result by composition. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and Commutative diagrams play the role in category theory that equations play in algebra. An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity.
Contents |
Examples
The first isomorphism theorem is a commutative triangle as follows:
Since 
, the left diagram is commutative; and since 
, so is the right diagram. In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural
Similarly, the square above is commutative if 
.
Symbols
In algebra texts, the type of morphism can be denoted with different arrow usages: monomorphisms with a 
, epimorphisms as a 
, and isomorphisms as a 
. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In the context of Abstract algebra or Universal algebra, a monomorphism is simply an Injective Homomorphism. In Category theory an epimorphism (also called an epic morphism or an epi) is a Morphism f: X &rarr Y which In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective This is common enough that texts often do not footnote explanations for the different arrows.
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. In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit
Diagram chasing
Diagram chasing is a method of mathematical proof used especially in homological algebra. In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting 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. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every In Mathematics, especially in Homological algebra and other applications of Abelian category theory as well as in Differential geometry and Group A syllogism is constructed, for which the graphical display of the diagram is just a visual aid. A syllogism, or logical appeal, (συλλογισμός &mdash "conclusion" "inference" (usually the categorical syllogism) is a kind of 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. In Mathematics, especially Homological algebra and other applications of Abelian category theory the five lemma is an important and widely used lemma In Mathematics, particularly Homological algebra, the snake lemma, a statement valid in every Abelian category, is the crucial tool used to construct the In Mathematics, the nine lemma is a statement about Commutative diagrams and Exact sequences valid in any Abelian category, as well as in the category
External links
MathWorld is an online Mathematics reference work created and largely written by Eric WDapyx Software network: MP3 Explorer | Ebook Manager | Zenithic