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| 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 [[morphism]]s that also need indexing. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a ''function'' from a fixed index ''set'' to the class of ''sets''. A diagram is a collection of objects and morphisms, indexed by a fixed category; equivalently, a ''functor'' from a fixed index ''category'' to some ''category''.
| | Greetings! I am Myrtle Shroyer. California is where her home std test, [http://inspirationpedi.com/groups/useful-guidance-for-battling-your-yeast-infection/ Get More], is but she needs to transfer because of her family members. My working day job is a meter reader. What I love doing is taking part in baseball but I haven't made a dime with it. |
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| Diagrams are central to the definition of [[limit (category theory)|limits and colimits]], and to the related notion of [[cone (category theory)|cone]]s.
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| ==Definition==
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| Formally, a '''diagram''' of type ''J'' in a [[category (mathematics)|category]] ''C'' is a ([[Covariance and contravariance of functors|covariant]]) [[functor]]
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| :''D'' : ''J'' → ''C''
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| The category ''J'' is called the '''index category''' or the '''scheme''' of the diagram ''D''; the functor is sometimes called a '''''J''-shaped diagram'''.<ref>J.P. May, ''A Concise Course in Algebraic Topology'', (1999) The University of Chicago Press, ISBN 0-226-51183-9</ref> The actual objects and morphisms in ''J'' are largely irrelevant, only the way in which they are interrelated matters. The diagram ''D'' is thought of as indexing a collection of objects and morphisms in ''C'' patterned on ''J''.
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| Although, technically, there is no difference between an individual ''diagram'' and a ''functor'' or between a ''scheme'' and a ''category'', the change in terminology reflects a change in perspective, just as in the set theoretic case: one fixes the index category, and allows the functor (and, secondarily, the target category) to vary.
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| One is most often interested in the case where the scheme ''J'' is a [[small category|small]] or even [[Finite set|finite]] category. A diagram is said to be '''small''' or '''finite''' whenever ''J'' is.
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| A morphism of diagrams of type ''J'' in a category ''C'' is a [[natural transformation]] between functors. One can then interpret the '''category of diagrams''' of type ''J'' in ''C'' as the [[functor category]] ''C''<sup>''J''</sup>, and a diagram is then an object in this category.
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| ==Examples==
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| * Given any object ''A'' in ''C'', one has the '''constant diagram''', which is the diagram that maps all objects in ''J'' to ''A'', and all morphisms of ''J'' to the identity morphism on ''A''. Notationally, one often uses an underbar to denote the constant diagram: thus, for any object <math>A</math> in ''C'', one has the constant diagram <math>\underline A</math>.
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| * If ''J'' is a (small) [[discrete category]], then a diagram of type ''J'' is essentially just an [[indexed family]] of objects in ''C'' (indexed by ''J''). When used in the construction of the [[limit (category theory)|limit]], the result is the [[product (category theory)|product]]; for the colimit, one gets the [[coproduct]]. So, for example, when ''J'' is the discrete category with two objects, the resulting limit is just the binary product.
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| * If ''J'' = -1 ← 0 → +1, then a diagram of type ''J'' (''A'' ← ''B'' → ''C'') is a [[span (category theory)|span]], and its colimit is a [[Pushout (category theory)|pushout]]. If one were to "forget" that the diagram had object ''B'' and the two arrows ''B'' → ''A'', ''B'' → ''C'', the resulting diagram would simply be the discrete category with the two objects ''A'' and ''C'', and the colimit would simply be the binary coproduct. Thus, this example shows an important way in which the idea of the diagram generalizes that of the [[index set]] in set theory: by including the morphisms ''B'' → ''A'', ''B'' → ''C'', one discovers additional structure in constructions built from the diagram, structure that would not be evident if one only had an index set with no relations between the objects in the index.
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| * If ''J'' = -1 → 0 ← +1, then a diagram of type ''J'' (''A'' → ''B'' ← ''C'') is a [[cospan]], and its limit is a [[Pullback (category theory)|pullback]].
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| * The index <math>J = 0 \overrightarrow{\to} 1</math> is called "two parallel morphisms", or sometimes the [[free quiver]] or the [[walking quiver]]. A diagram of type ''J'' (<math>f,g\colon X \to Y</math>) is then a [[quiver (mathematics)|quiver]]; its limit is an [[Equaliser (mathematics)|equalizer]], and its colimit is a [[coequalizer]].
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| * If ''J'' is a [[poset category]], then a diagram of type ''J'' is a family of objects ''D''<sub>''i''</sub> together with a unique morphism ''f''<sub>''ij''</sub> : ''D''<sub>''i''</sub> → ''D''<sub>''j''</sub> whenever ''i'' ≤ ''j''. If ''J'' is [[directed set|directed]] then a diagram of type ''J'' is called a [[direct system (mathematics)|direct system]] of objects and morphisms. If the diagram is [[contravariant functor|contravariant]] then it is called an [[inverse system]].
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| ==Cones and limits==
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| A [[cone (category theory)|cone]] with vertex ''N'' of a diagram ''D'' : ''J'' → ''C'' is a morphism from the constant diagram Δ(''N'') to ''D''. The constant diagram is the diagram which sends every object of ''J'' to an object ''N'' of ''C'' and every morphism to the identity morphism on ''N''.
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| The [[limit (category theory)|limit]] of a diagram ''D'' is a [[universal cone]] to ''D''. That is, a cone through which all other cones uniquely factor. If the limit exists in a category ''C'' for all diagrams of type ''J'' one obtains a functor
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| :lim : ''C''<sup>''J''</sup> → ''C''
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| which sends each diagram to its limit.
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| Dually, the [[colimit]] of diagram ''D'' is a universal cone from ''D''. If the colimit exists for all diagrams of type ''J'' one has a functor
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| :colim : ''C''<sup>''J''</sup> → ''C''
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| which sends each diagram to its colimit.
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| == Commutative diagrams ==
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| {{main|Commutative diagram}}
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| Diagrams and functor categories are often visualized by [[commutative diagrams]], particularly if the index category is a finite [[poset category]] with few elements: one draws a commutative diagram with a node for every object in the index category, and an arrow for a generating set of morphisms, omitting identity maps and morphisms that can be expressed as compositions. The commutativity corresponds to the uniqueness of a map between two objects in a poset category. Conversely, every commutative diagram represents a diagram (a functor from a poset index category) in this way.
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| Not every diagram commutes, as not every index category is a poset category:
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| 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 \overrightarrow{\to} \bullet</math>; <math>f,g\colon X \to Y</math>) need not commute. Further, diagrams may be impossible to draw (because infinite) or simply messy (because too many objects or morphisms); however, schematic commutative diagrams (for subcategories of the index category, or with ellipses, such as for a directed system) are used to clarify such complex diagrams.
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| == See also ==
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| * [[Direct system (mathematics)|Direct system]]
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| * [[Inverse system]]
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| ==References==
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| {{reflist}}
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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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| * {{nlab|id=diagram}}
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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, commutative diagrams, categories, [[functor]]s, [[natural transformation]]s.
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| [[Category:Functors]]
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Greetings! I am Myrtle Shroyer. California is where her home std test, Get More, is but she needs to transfer because of her family members. My working day job is a meter reader. What I love doing is taking part in baseball but I haven't made a dime with it.