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| In [[category theory]], a branch of [[mathematics]], the '''opposite category''' or '''dual category''' ''C''<sup>op</sup> of a given category ''C'' is formed by reversing the [[morphism]]s, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite category is the original category itself. In symbols, <math>(C^{op})^{op} = C</math>.
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| ==Examples==
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| * An example comes from reversing the direction of inequalities in a [[partial order]]. So if ''X'' is a [[Set (mathematics)|set]] and ≤ a partial order relation, we can define a new partial order relation ≤<sub>new</sub> by
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| :: ''x'' ≤<sub>new</sub> ''y'' if and only if ''y'' ≤ ''x''.
| | Individual who wrote the statement is [http://www.google.co.uk/search?hl=en&gl=us&tbm=nws&q=called+Roberto&gs_l=news called Roberto] Ledbetter and his wife doesn't like it at nearly all. In his professional life he can be a people manager. He's always loved living to Guam and he has everything that he conditions there. The preference hobby for him and as a result his kids is farming but he's been removing on new things lately. He's been working on his website for some spare time now. Check it out here: http://prometeu.net<br><br>My web blog; [http://prometeu.net hack clash of clans ipad] |
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| : For example, there are opposite pairs child/parent, or descendant/ancestor.
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| * The category of [[Boolean algebra (structure)|Boolean algebra]]s and Boolean [[homomorphism]]s is [[Equivalence of categories|equivalent]] to the opposite of the category of [[Stone space]]s and continuous functions.
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| * The category of [[affine scheme]]s is [[equivalence (category theory)|equivalent]] to the opposite of the category of [[commutative ring]]s.
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| * The [[Pontryagin duality]] restricts to an equivalence between the category of [[Compact_space#Definition|compact]] [[Hausdorff space|Hausdorff]] [[abelian group|abelian]] [[topological group]]s and the opposite of the category of (discrete) abelian groups.
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| * By the Gelfand-Neumark theorem, the category of localizable [[Sigma-algebra|measurable spaces]] (with [[measurable function|measurable maps]]) is equivalent to the category of commutative [[Von Neumann algebra]]s (with [[Normal operator|normal]] [[Unital map|unital]] homomorphisms of *-algebras).<ref name=MO1>{{cite web|url=http://mathoverflow.net/questions/20740/is-there-an-introduction-to-probability-theory-from-a-structuralist-categorical-p|title=Is there an introduction to probability theory from a structuralist/categorical perspective?|publisher=MathOverflow|accessdate=25 October 2010}}</ref>
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| ==Properties== | |
| Opposite preserves products:
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| :<math>(C\times D)^{op} \cong C^{op}\times D^{op}</math> (see [[product category]])
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| Opposite preserves [[functors]]:
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| :<math>(\mathrm{Funct}(C,D))^{op} \cong \mathrm{Funct}(C^{op},D^{op})</math><ref>H. Herrlich, G. E. Strecker, ''Category Theory'', 3rd Edition, Heldermann Verlag, p. 99.</ref><ref>O. Wyler, ''Lecture Notes on Topoi and Quasitopoi'', World Scientific, 1991, p. 8.</ref> (see [[functor category]], [[opposite functor]])
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| Opposite preserves slices:
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| :<math>(F\downarrow G)^{op} \cong (G^{op}\downarrow F^{op})</math> (see [[comma category]]) | |
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| ==See also==
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| * [[Dual object]]
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| * [[Dual (category theory)]]
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| * [[Duality (mathematics)]]
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| * [[Contravariant functor]]
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| * [[Opposite functor]]
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| ==References==
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| {{Reflist}}
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| * {{nlab|id=opposite+category|title=Opposite category}}
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| {{DEFAULTSORT:Opposite Category}}
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| [[Category:Category theory]]
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