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Commutative diagram

 

 
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Commutative diagram
 
 
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In mathematics, and especially in category theory
Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
 a commutative diagram is a diagram
Diagram

A diagram is a 2D geometric model symbolic representation of information according to some visualization technique. Sometimes, the technique uses a Three-dimensional space visualization which is then graphical projection onto the 2D surface....
 of objects, also known as vertices, and morphism
Morphism

In mathematics, a morphism is an Abstraction derived from structure-preserving map between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory....
s, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition
Composition

Composition can refer to:* fallacy of composition, a fallacy of ambiguation in which one assumes that a whole has a property solely because its various parts have that property...
. Commutative diagrams play the role in category theory that equations play in algebra
Algebra

Algebra is a branch of mathematics concerning the study of structure , relation , and quantity. Together with geometry, mathematical analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics....
.

he following diagram expressing the first isomorphism theorem, commutativity means that :

Below is a generic commutative square, in which
lgebra texts, the type of morphism
Morphism

In mathematics, a morphism is an Abstraction derived from structure-preserving map between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory....
 can be denoted with different arrow usages: monomorphism
Monomorphism

In the context of abstract algebra or universal algebra, a monomorphism is an Injective function homomorphism. A monomorphism from X to Y is often denoted with the notation ....
s with a , epimorphism
Epimorphism

In category theory an epimorphism is a morphism f : X ? Y which is Cancellation property in the sense that, for all morphisms ,Epimorphisms are analogues of surjective functions, but they are not exactly the same....
s with a , and isomorphism
Isomorphism

In abstract algebra, an isomorphism is a bijection map f such that both f and its inverse function f −1 are homomorphisms, i.e., structure-preserving mappings....
s with a .






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In mathematics, and especially in category theory
Category theory

In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them: it abstracts from set s and function s to objects linked in diagrams by morphisms or arrows....
 a commutative diagram is a diagram
Diagram

A diagram is a 2D geometric model symbolic representation of information according to some visualization technique. Sometimes, the technique uses a Three-dimensional space visualization which is then graphical projection onto the 2D surface....
 of objects, also known as vertices, and morphism
Morphism

In mathematics, a morphism is an Abstraction derived from structure-preserving map between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory....
s, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition
Composition

Composition can refer to:* fallacy of composition, a fallacy of ambiguation in which one assumes that a whole has a property solely because its various parts have that property...
. Commutative diagrams play the role in category theory that equations play in algebra
Algebra

Algebra is a branch of mathematics concerning the study of structure , relation , and quantity. Together with geometry, mathematical analysis, combinatorics, and number theory, algebra is one of the main branches of mathematics....
.

Examples

In the following diagram expressing the first isomorphism theorem, commutativity means that :

Below is a generic commutative square, in which

Symbols

In algebra texts, the type of morphism
Morphism

In mathematics, a morphism is an Abstraction derived from structure-preserving map between two mathematical structures.The study of morphisms and of the structures over which they are defined, is central to category theory....
 can be denoted with different arrow usages: monomorphism
Monomorphism

In the context of abstract algebra or universal algebra, a monomorphism is an Injective function homomorphism. A monomorphism from X to Y is often denoted with the notation ....
s with a , epimorphism
Epimorphism

In category theory an epimorphism is a morphism f : X ? Y which is Cancellation property in the sense that, for all morphisms ,Epimorphisms are analogues of surjective functions, but they are not exactly the same....
s with a , and isomorphism
Isomorphism

In abstract algebra, an isomorphism is a bijection map f such that both f and its inverse function f −1 are homomorphisms, i.e., structure-preserving mappings....
s with a . The dashed arrow typically means that this diagram claims that the indicated morphism exists. This is common enough that texts often do not explain the meanings of the different types of arrow.

Verifying commutativity

Commutativity makes sense for a polygon
Polygon

In geometry a polygon is traditionally a plane Shape that is bounded by a closed curve path or circuit, composed of a finite sequence of straight line segments ....
 of any finite number of sides (including just 1 or 2), and a diagram is commutative if every polygonal subdiagram is commutative.

Diagram chasing

Diagram chasing is a method of mathematical proof
Mathematical proof

In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive reasoning or empirical arguments....
 used especially in homological algebra
Homological algebra

Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincar? and David Hilbert....
. Given a commutative diagram, a proof by diagram chasing involves the formal use of the properties of the diagram, such as injective or surjective maps, or exact sequence
Exact sequence

In mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory, an exact sequence is a sequence of objects and morphisms between them such that the of one morphism equals the kernel of the next....
s. A syllogism
Syllogism

A syllogism, or logical appeal, , is a kind of logical argument in which one proposition is Inference from two others of a certain form....
 is constructed, for which the graphical display of the diagram is just a visual aid. It follows that 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
Five lemma

In mathematics, especially homological algebra and other applications of Abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams....
, the snake lemma
Snake lemma

In mathematics, particularly homological algebra, the snake lemma, a statement valid in every abelian category, is the crucial tool used to construct the long exact sequences that are ubiquitous in homological algebra and its applications, for instance in algebraic topology....
, the zig-zag lemma
Zig-zag lemma

In mathematics, particularly homological algebra, the zig-zag lemma asserts the existence of a particular long exact sequence in the homology groups of certain chain complexes....
, and the nine lemma
Nine lemma

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 of group s....
.

Diagrams as functors


A commutative diagram in a category C can be interpreted as a functor
Functor

In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of small categories....
 from an index category J to C; one calls the functor a diagram
Diagram (category theory)

In category theory, a branch of mathematics, a diagram is the categorical analogue of a indexed family in set theory. The primary difference is that in the categorical setting one has morphisms as well: an indexed family of sets is a collection of sets, indexed by a fixed set , while a diagram is a collection of objects and morphisms, indexed...
.

More formally, a commutative diagram is a visualization of a diagram indexed by a poset category:
  • one draws a node for every object in the index category,
  • an arrow for a generating set of morphisms,
    omitting identity maps and morphisms that can be expressed as compositions,
  • and the commutativity of the diagram (the equality of different compositions of maps between two objects) corresponds to the uniqueness of a map between two objects in a poset category.


Conversely, given a commutative diagram, it defines a poset category:
  • the objects are the nodes,
  • there is a morphism between any two objects if and only if there is a (directed) path between the nodes,
  • with the relation that this morphism is unique (any composition of maps is defined by its domain and target: this is the commutativity axiom).


However, not every diagram commutes (the notion of diagram strictly generalizes commutative diagram): most simply, the diagram of a single object with an endomorphism , or with two parallel arrows (; ), as used in the definition of equalizer need not commute. Further, diagrams may be impossible (because infinite) or messy (because many objects or morphisms) to draw.

See also

  • Diagram (category theory)
    Diagram (category theory)

    In category theory, a branch of mathematics, a diagram is the categorical analogue of a indexed family in set theory. The primary difference is that in the categorical setting one has morphisms as well: an indexed family of sets is a collection of sets, indexed by a fixed set , while a diagram is a collection of objects and morphisms, indexed...
  • Mathematical diagram
    Mathematical diagram

    Mathematical diagrams are diagrams in the field of mathematics, and diagrams using mathematics such as charts and graphs, that are mainly designed to convey mathematical relationships, for example, comparisons over time....


External links

  • at MathWorld
    MathWorld

    MathWorld is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by Wolfram Research Inc. and was partially funded by the National Science Foundation's National Science Digital Library grant to the University of Illinois at Urbana-Champaign....