Inmathematics, and especially incategory theory, acommutative diagram is adiagram such that all directed paths in the diagram with the same start and endpoints lead to the same result.[1] It is said that commutative diagrams play the role in category theory thatequations play inalgebra.[2]
A commutative diagram often consists of three parts:
In algebra texts, the type of morphism can be denoted with different arrow usages:
The meanings of different arrows are not entirely standardized: the arrows used for monomorphisms, epimorphisms, and isomorphisms are also used forinjections,surjections, andbijections, as well as the cofibrations, fibrations, and weak equivalences in amodel category.
Commutativity makes sense for apolygon 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.
In the left diagram, which expresses thefirst isomorphism theorem, commutativity of the triangle means that. In the right diagram, commutativity of the square means.
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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.
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Diagram chasing (also calleddiagrammatic search) is a method ofmathematical proof used especially inhomological algebra, 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 asinjective orsurjective maps, orexact sequences.[5] Asyllogism 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 thefive lemma, thesnake lemma, thezig-zag lemma, and thenine lemma.
In higher category theory, one considers not only objects and arrows, but arrows between the arrows, arrows between arrows between arrows, and so onad infinitum. For example, the category of small categoriesCat is naturally a 2-category, withfunctors as its arrows andnatural 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 categoriesC andD, together with two functorsF,G :C →D and a natural transformationα :F ⇒G:
There are two kinds of composition in a 2-category (calledvertical composition andhorizontal composition), and they may also be depicted viapasting diagrams (see2-category#Definition for examples).
A commutative diagram in a categoryC can be interpreted as afunctor from an index categoryJ toC; one calls the functor adiagram.
More formally, a commutative diagram is a visualization of a diagram indexed by aposet category. Such a diagram typically includes:
Conversely, given a commutative diagram, it defines a poset category, where:
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 thefree quiver), as used in the definition ofequalizer need not commute. Further, diagrams may be messy or impossible to draw, when the number of objects or morphisms is large (or even infinite).
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