350px, The commutative diagram used in the proof of the ">five lemma
In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, and especially in
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
, 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 o ...
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 play in
algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
.
Description
A commutative diagram often consists of three parts:
*
objects (also known as ''vertices'')
*
morphisms (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 may be labeled with a
or a
.
* An
epimorphism may be labeled with a
.
* An
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
may be labeled with a
.
* 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
.
** If the morphism is in addition unique, then the dashed arrow may be labeled
or
.
*If the morphism acts between two arrows (such as in the case of
higher category theory
In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit morphism, arrows in order to be able to explicitly study the structure behind those equalities. H ...
), it's called preferably a
natural transformation
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 categories involved. Hence, a natur ...
and may be labelled as
(as seen below in this article).
The meanings of different arrows are not entirely standardized: the arrows used for monomorphisms, epimorphisms, and isomorphisms are also used for
injections,
surjections, and
bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
s, as well as the cofibrations, fibrations, and weak equivalences in a
model category.
Verifying commutativity
Commutativity makes sense for a
polygon
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain.
The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the 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.
Examples
Example 1
In the left diagram, which expresses the
first isomorphism theorem, commutativity of the triangle means that
. In the right diagram, commutativity of the square means
.
Example 2
In order for the diagram below to commute, three equalities must be satisfied:
#
#
#
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 a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use othe ...
used especially in
homological algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
, 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, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
or
surjective
In mathematics, a surjective function (also known as surjection, or onto function ) is a function such that, for every element of the function's codomain, there exists one element in the function's domain such that . In other words, for a f ...
maps, or
exact sequences.
A
syllogism
A syllogism (, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true.
In its earliest form (defin ...
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
zig-zag lemma, and the
nine lemma.
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. For example, the category of small categories Cat is naturally a 2-category, with
functors as its arrows and
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:
. 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 (see
2-category#Definitions for examples).
Diagrams as functors
A commutative diagram in a category ''C'' can be interpreted as a
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
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 o ...
.
More formally, a commutative diagram is a visualization of a diagram indexed by a
poset category. Such a diagram typically includes:
* 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 (
), or with two parallel arrows (
, that is,
, sometimes called the
free quiver), 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
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 Chasingat
MathWorld
WildCatsis a category theory package for
Mathematica. Manipulation and visualization of objects,
morphisms, categories,
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
s,
natural transformation
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 categories involved. Hence, a natur ...
s.
{{Category theory
Homological algebra
Category theory
Mathematical proofs
Mathematical terminology
Diagrams