Commutative diagram

# Commutative diagram

Discussion

Encyclopedia
In mathematics, and especially in category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

, a commutative diagram is a diagram
Diagram (category theory)
In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a function...

of objects (also known as vertices) and morphism
Morphism
In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

s (also known as arrows or edges) such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition. Commutative diagrams play the role in category theory that equations play in algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...

.

Note that a diagram may not be commutative, i.e. the composition of different paths in the diagram may not give the same result. For clarification, phrases like "this commutative diagram" or "the diagram commutes" may be used.

## 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 mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

can be denoted with different arrow usages: monomorphism
Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from X to Y is often denoted with the notation X \hookrightarrow Y....

s with a , epimorphism
Epimorphism
In category theory, an epimorphism is a morphism f : X → Y which is right-cancellative in the sense that, for all morphisms ,...

s with a , and isomorphism
Isomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

s with a . The dashed arrow typically represents the claim that the indicated morphism exists whenever the rest of the diagram holds. 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 a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, 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 or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...

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 Poincaré and...

. 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
An exact sequence is a concept in mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory...

s. A syllogism
Syllogism
A syllogism is a kind of logical argument in which one proposition is inferred from two or more 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
The snake lemma is a tool used in mathematics, particularly homological algebra, to construct long exact sequences. The snake lemma is valid in every abelian category and is a crucial tool 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 groups. It states: ifis a commutative diagram and all columns as well as the two bottom rows are exact, then the top row is exact as well...

.

## 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 homomorphisms between categories, or morphisms when 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 an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a function...

.

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 messy or impossible to draw when the number of objects or morphisms is large (or even infinite).