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| [[image:FiveLemma.png|350px|thumb|The commutative diagram used in the proof of the [[five lemma]].]]
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| In mathematics, and especially in [[category theory]], a '''commutative diagram''' is a [[Diagram (category theory)|diagram]] of objects (also known as ''vertices'') and [[morphism]]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 [[category (mathematics)#Definition|composition]]. Commutative diagrams play the role in category theory that [[equations]] play in [[algebra]] (see Barr-Wells, Section 1.7).
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| 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.
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| ==Examples==
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| In the following diagram expressing the [[first isomorphism theorem]], commutativity means that <math>f = \tilde{f} \circ \pi</math>:
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| [[Image:First isomorphism theorem (plain).svg|175px]]
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| Below is a generic commutative square, in which <math>h \circ f = k \circ g</math>
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| [[Image:Commutative square.svg|150px]]
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| ===Symbols===
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| In algebra texts, the type of [[morphism]] can be denoted with different arrow usages: [[monomorphism]]s with a <math>\hookrightarrow</math>, [[epimorphism]]s with a <math>\twoheadrightarrow</math>, and [[isomorphism]]s with a <math>\overset{\sim}{\rightarrow}</math>. 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.
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| ==Verifying commutativity==
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| Commutativity makes sense for a [[polygon]] of any finite number of sides (including just 1 or 2), and a diagram is commutative if every polygonal subdiagram is commutative.
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| ==Diagram chasing==
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| '''Diagram chasing''' is a method of [[mathematical proof]] used especially in [[homological algebra]]. 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]]s. A [[syllogism]] 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.
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| 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]].
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| == Diagrams as functors ==
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| {{Main|Diagram (category theory)}}
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| A commutative diagram in a category ''C'' can be interpreted as a [[functor]] from an index category ''J'' to ''C;'' one calls the functor a '''[[diagram (category theory)|diagram]].'''
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| More formally, a commutative diagram is a visualization of a diagram indexed by a [[poset category]]:
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| * one draws a node for every object in the index category,
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| * an arrow for a generating set of morphisms,
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| *:omitting identity maps and morphisms that can be expressed as compositions,
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| * 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.
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| Conversely, given a commutative diagram, it defines a poset category:
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| * the objects are the nodes,
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| * there is a morphism between any two objects if and only if there is a (directed) path between the nodes,
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| * with the relation that this morphism is unique (any composition of maps is defined by its domain and target: this is the commutativity axiom).
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| However, not every diagram commutes (the notion of diagram strictly generalizes commutative diagram): most simply, the diagram of a single object with an endomorphism (<math>f\colon X \to X</math>), or with two parallel arrows (<math>\bullet \rightrightarrows \bullet</math>, that is, <math>f,g\colon X \to Y</math>, sometimes called the [[free quiver]]), as used in the definition of [[equaliser (mathematics)|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).
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| == See also ==
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| * [[Mathematical diagram]]
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| ==References==
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| *{{cite book | last = Adámek | first = Jiří | coauthors = Horst Herrlich, and George E. Strecker | year = 1990 | url =http://katmat.math.uni-bremen.de/acc/acc.pdf | title = Abstract and Concrete Categories | publisher = John Wiley & Sons | isbn = 0-471-60922-6}} Now available as free on-line edition (4.2MB PDF).
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| * {{Cite book| last1=Barr| first1=Michael|authorlink1=Michael Barr (mathematician) | last2=Wells| first2=Charles| authorlink2=Charles Wells (mathematician) | year=2002| title=Toposes, Triples and Theories|url=http://www.tac.mta.ca/tac/reprints/articles/12/tr12.pdf |isbn=0-387-96115-1}} Revised and corrected free online version of ''Grundlehren der mathematischen Wissenschaften (278)'' Springer-Verlag, 1983).
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| == External links ==
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| * [http://mathworld.wolfram.com/DiagramChasing.html Diagram Chasing] at [[MathWorld]]
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| * [http://wildcatsformma.wordpress.com WildCats] is a category theory package for [[Mathematica]]. Manipulation and visualization of objects, [[morphism]]s, categories, [[functor]]s, [[natural transformation]]s.
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| [[Category:Homological algebra]]
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| [[Category:Category theory]]
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| [[Category:Mathematical proofs]]
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| [[Category:Mathematical terminology]]
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| [[Category:Diagrams]]
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