en>AnomieBOT: Dating maintenance tags: {{Refimprove}}

2014-09-18T19:19:02Z

Dating maintenance tags: {{Refimprove}}

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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>.

	~~==Examples==~~
	* 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

	~~:: ''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]

	~~: For example, there are opposite pairs child/parent, or descendant/ancestor.~~

	* 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.

	* The category of [[affine scheme]]s is [[equivalence (category theory)\|equivalent]] to the opposite of the category of [[commutative ring]]s.

	* 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.~~

	* 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>~~

	==~~Properties=~~=
	~~Opposite preserves products:~~
	~~:<math>(C\times D)^{op} \cong C^{op}\times D^{op}</math> (see [[product category~~]])

	~~Opposite preserves [[functors]]:~~
	~~:<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]])~~

	~~Opposite preserves slices~~:
	:<~~math~~>~~(F\downarrow G)^{op} \cong (G^{op}\downarrow F^{op})~~<~~/math~~> ~~(see~~ [~~[comma category]])~~

	~~==See also==~~
	* [[Dual object]]
	* [[Dual (category theory)]]
	* [[Duality (mathematics)]]
	* [[Contravariant functor]]
	* [[Opposite functor]]

	~~==References==~~
	~~{{Reflist}}~~

	* {{nlab\|id=opposite+category\|title=Opposite category}}

	~~{{DEFAULTSORT~~:~~Opposite Category}}~~
	~~[[Category:Category theory]~~]

en>Maxal: /* External links */

2013-12-15T01:21:32Z

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en>Arthur Rubin: Reverted good faith edits by Richard.therooster.john (talk): I'm opposed to the link, but, even if it were appropriate, an external link to a Wikipe...

2012-04-30T15:45:24Z

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en>AnomieBOT: Dating maintenance tags: {{Refimprove}}

en>Maxal: /* External links */

en>Arthur Rubin: Reverted good faith edits by Richard.therooster.john (talk): I'm opposed to the link, but, even if it were appropriate, an external link to a Wikipe...