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350px, The commutative diagram used in the proof of the five lemma. In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, and especially in
category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...
, a commutative diagram is a
diagram A diagram is a symbolic Depiction, representation of information using Visualization (graphics), visualization techniques. Diagrams have been used since prehistoric times on Cave painting, walls of caves, but became more prevalent during the Age ...
such that all directed paths in the diagram with the same start and endpoints lead to the same result. It is said that commutative diagrams play the role in category theory that
equations In mathematics, an equation is a statement that asserts the equality (mathematics), equality of two Expression (mathematics), expressions, which are connected by the equals sign "=". The word ''equation'' and its cognates in other languages may ...
play in
algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

algebra
(see ).


Description

A commutative diagram often consists of three parts: *
objects Object may refer to: General meanings * Object (philosophy) An object is a philosophy, philosophical term often used in contrast to the term ''Subject (philosophy), subject''. A subject is an observer and an object is a thing observed. For mo ...
(also known as ''vertices'') *
morphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

morphism
s (also known as ''arrows'' or ''edges'') * paths or composites


Arrow symbols

In algebra texts, the type of morphism can be denoted with different arrow usages: * A
monomorphism In the context of abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, rin ...
(injective homomorphism) may be labeled with a \hookrightarrow. * An
epimorphism In category theory, an epimorphism (also called an epic morphism or, colloquially, an epi) is a morphism ''f'' : ''X'' → ''Y'' that is Cancellation property, right-cancellative in the sense that, for all objects ''Z'' and all morphisms , : g_1 ...
(surjective homomorphism) may be labeled with a \twoheadrightarrow. * An
isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

isomorphism
(bijective homomorphism) may be labeled with a \overset. * The dashed arrow typically represents the claim that the indicated morphism exists (whenever the rest of the diagram holds); the arrow may be optionally labeled as \exists. ** If the morphism is in addition unique, then the dashed arrow may be labeled ! or \exists!.


Verifying commutativity

Commutativity makes sense for a
polygon In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position o ...

polygon
of any finite number of sides (including just 1 or 2), and a diagram is commutative if every polygonal subdiagram is commutative. Note that a diagram may be non-commutative, i.e., the composition of different paths in the diagram may not give the same result.


Phrases

Phrases like "this commutative diagram" or "the diagram commutes" may be used.


Examples

In the left diagram, which expresses the
first isomorphism theorem In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between Quotient (universal algebra), quotients, homomorphisms, and subobjects. Vers ...

first isomorphism theorem
, commutativity of the triangle means that f = \tilde \circ \pi. In the right diagram, commutativity of the square means h \circ f = k \circ g. In order for the diagram below to commute, three equalities must be satisfied: # r \circ h \circ g = H \circ G \circ l # m \circ g = G \circ l # r \circ h = H \circ m Here, since the first equality follows from the last two, it suffices to show that (2) and (3) are true in order for the diagram to commute. However, since equality (3) generally does not follow from the other two, it is generally not enough to have only equalities (1) and (2) if one were to show that the diagram commutes.


Diagram chasing

Diagram chasing (also called diagrammatic search) is a method of
mathematical proof A mathematical proof is an inferential argument In logic Logic is an interdisciplinary field which studies truth and reasoning Reason is the capacity of consciously making sense of things, applying logic Logic (from Ancient Greek ...
used especially in
homological algebra Homological algebra is the branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contain ...
, where one establishes a property of some morphism by tracing the elements of a commutative diagram. A proof by diagram chasing typically involves the formal use of the properties of the diagram, such as
injective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
or
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
maps, or
exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same ...
s. A
syllogism A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument In logic and philosophy, an argument is a series of statements (in a natural language), called the premises or premisses (bo ...
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, the
snake lemma The snake lemma is a tool used in mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their cha ...
, the
zig-zag lemmaIn 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. The result is valid in every abelian category. Statement In an abelia ...
, and the .


In higher category theory

In higher category theory, one considers not only objects and arrows, but arrows between the arrows, arrows between arrows between arrows, and so on
ad infinitum ''Ad infinitum'' is a Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of ...
. For example, the category of small categories Cat is naturally a 2-category, with
functors In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

functors
as its arrows and
natural transformations In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the Category (mathematics), categories inv ...

natural transformations
as the arrows between functors. In this setting, commutative diagrams may include these higher arrows as well, which are often depicted in the following style: \Rightarrow. For example, the following (somewhat trivial) diagram depicts two categories and , together with two functors , : → and a natural transformation : ⇒ : : There are two kinds of composition in a 2-category (called vertical composition and horizontal composition), and they may also be depicted via
pasting diagrams Paste is a term for any very thick viscous fluid. It may refer to: Science and technology * Adhesive or paste ** Wallpaper paste ** Wheatpaste, a liquid adhesive made from vegetable starch and water * Paste (rheology), a substance that behaves a ...
(see for examples).


Diagrams as functors

A commutative diagram in a category ''C'' can be interpreted as a
functor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

functor
from an index category ''J'' to ''C;'' one calls the functor a
diagram A diagram is a symbolic Depiction, representation of information using Visualization (graphics), visualization techniques. Diagrams have been used since prehistoric times on Cave painting, walls of caves, but became more prevalent during the Age ...
. More formally, a commutative diagram is a visualization of a diagram indexed by a
poset category In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...
. Such a diagram typically include: * 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), * the commutativity of the diagram (the equality of different compositions of maps between two objects), corresponding to the uniqueness of a map between two objects in a poset category. Conversely, given a commutative diagram, it defines a poset category, where: * 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). As a simple example, the diagram of a single object with an endomorphism (f\colon X \to X), or with two parallel arrows (\bullet \rightrightarrows \bullet, that is, f,g\colon X \to Y, sometimes called the
free quiver In mathematics, a quiver is a directed graph where loops and multiple arrows between two vertex (graph theory), vertices are allowed, i.e. a multidigraph. They are commonly used in representation theory: a representation ''V'' of a quiver assig ...
), 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).


See also

*
Mathematical diagram Mathematical diagrams, such as chart A chart is a graphical representation Graphic communication as the name suggests is communication using graphic elements. These elements include symbols such as glyphs and icon (computing), icons, images s ...


References


Bibliography

* Now available as free on-line edition (4.2MB PDF). * Revised and corrected free online version of ''Grundlehren der mathematischen Wissenschaften (278)'' Springer-Verlag, 1983).


External links


Diagram Chasing
at
MathWorld ''MathWorld'' is an online mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geome ...

WildCats
is a category theory package for
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning Machine learning (ML) is the study of computer algorithms that can improve automatically through experi ...

Mathematica
. Manipulation and visualization of objects,
morphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

morphism
s, categories,
functor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

functor
s,
natural transformation In category theory Category theory formalizes mathematical structure In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), s ...

natural transformation
s. {{Category theory Homological algebra Category theory Mathematical proofs Mathematical terminology Diagrams