Lebesgue's decomposition theorem - Revision history

en>Tudor987 at 13:29, 5 July 2014

2014-07-05T13:29:47Z

← Older revision		Revision as of 15:29, 5 July 2014
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en>Spinningspark: Reverted edits by 188.154.99.222 (talk) to last version by Schmock

2014-02-04T16:01:52Z

Reverted edits by 188.154.99.222 (talk) to last version by Schmock

← Older revision		Revision as of 18:01, 4 February 2014
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	~~In [[category theory]]~~, ~~'''filtered categories''' generalize the notion of [[directed set]] understood as a category~~ (~~hence called a directed category; while some use directed category as a synonym for a filtered category). There is a dual notion of '''cofiltered''' category which will be recalled below.~~		I would like to introduce myself to you, I am Jayson Simcox but I online psychic readings ([http://myoceancounty.net/groups/apply-these-guidelines-when-gardening-and-grow/ http://myoceancounty.net/groups/apply-these-guidelines-when-gardening-and-grow]) don't like when people use my complete title. The favorite pastime for him and his children is style and he'll be beginning some thing else alongside with it. Alaska is exactly where I've always been living. Credit authorising is how she makes a living.<br><br>Feel free to surf to my web blog - [http://test.jeka-nn.ru/node/129 psychic phone] love psychic - [http://findyourflirt.net/index.php?m=member_profile&p=profile&id=117823 findyourflirt.net] -


	~~==Filtered categories==~~

	A [~~[Category (mathematics)\|category]] <math>J<~~/~~math> is '''filtered''' when~~
	* it is not empty,
	* for every two objects <math>j</~~math> and <math>j'<~~/~~math> in <math>J<~~/~~math> there exists an object <math>k</math>~~ and ~~two arrows <math>f:j\to k<~~/~~math> and <math>f'~~:~~j'\to k<~~/~~math> in <math>J<~~/~~math>,~~
	* for every two parallel arrows <math>u,v:i\to j</~~math> in <math>J<~~/~~math>, there exists an object <math>k</math>~~ and ~~an arrow <math>w:j\to k</math> such that <math>wu=wv</math>~~.

	~~A diagram~~ is ~~said to~~ be ~~of cardinality <math>\kappa</math> if the morphism set of its domain~~ is ~~of cardinality <math>\kappa</math>~~. ~~A category <math>J</math> is filtered if and only if there~~ is a ~~cone over any finite diagram~~ <~~math~~>~~d: D\to J~~<~~/math~~>~~; more generally, for a regular cardinal <math>\kappa</math>, a category <math>J</math> is said~~ to ~~be <math>\kappa</math>~~-~~filtered if for every diagram <math>d</math> in <math>J</math> of cardinality smaller than <math>\kappa</math> there is a cone over <math>d</math>.~~

	~~A '''filtered colimit''' is a~~ [~~[colimit]] of a [[functor]] <math>F~~:~~J\to C<~~/~~math> where <math>J<~~/~~math> is a filtered category~~. ~~This readily generalizes to <math>\kappa</math>~~-~~filtered limits~~. ~~An '''ind-object''' in a category <math>C</math> is a presheaf of sets <math>C^{op}\to Set<~~/~~math> which is a small filtered colimit of representable presheaves. Ind-objects in a category <math>C<~~/~~math> form a full subcategory <math>Ind(C)</math> in the category of functors <math>C^{op}\to Set</math>. The category <math>Pro(C)=Ind(C^{op})^{op}</math> of pro~~-~~objects in <math>C</math> is the opposite of the category of ind-objects in the opposite category <math>C^{op}</math>.~~

	~~==Cofiltered categories==~~
	~~A category <math>J</math> is cofiltered if the [~~[~~opposite category]] <math>J^{\mathrm{op}}</math> is filtered. In detail, a category is cofiltered when~~
	* it is not empty
	* for every two objects <math>j</math> and <math>j'</math> in <math>J</math> there exists an object <math>k</math> and two arrows <math>f:~~k\to j<~~/~~math> and <math>f':k \to j'<~~/~~math> in <math>J</math>,~~
	* for every two parallel arrows <math>u,v:j\to i</math> in <math>J</math>, there exists an object <math>k</math> and an arrow <math>w:k\to j</math> such that <math>uw=vw</math>.

	~~A '''cofiltered limit''' is a [[limit (category theory)\|limit]] of a [[functor]] <math>F:J \to C</math> where <math>J<~~/~~math> is a cofiltered category~~.

	==~~References~~==
	* Artin, M., Grothendieck, A. and Verdier, J. L. ''Séminaire de Géométrie Algébrique du Bois Marie (SGA 4)''. Lecture Notes in Mathematics 269, Springer Verlag, 1972. ~~Exposé I, 2.7.~~
	* {{Citation \| last1=Mac Lane \| first1=Saunders \| author1-link=Saunders Mac Lane \| title=[[Categories for the Working Mathematician]] ~~\| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| edition=2nd \| isbn=978~~-~~0-387-98403-2 \| year=1998}}, section IX.1.~~

	~~<references/>~~

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

188.154.99.222: /* References */

2014-01-28T13:11:21Z

References

← Older revision		Revision as of 15:11, 28 January 2014
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	~~The name of the writer is Numbers but it~~'~~s not~~ the ~~most masucline title out there~~. ~~My day occupation~~ is a ~~meter reader~~. ~~Puerto Rico~~ is ~~where he~~'~~s been living~~ for ~~years~~ and ~~he will by no means transfer~~. ~~The favorite hobby~~ for ~~my kids and me~~ is to ~~play baseball but I haven~~'~~t produced~~ a ~~dime with it~~.<br><br>~~Also visit my blog~~ - [~~http~~://~~www~~.~~eddysadventurestore.nl~~/~~nieuws~~/~~eliminate~~-~~candida~~-~~one~~-~~these~~-~~tips over the counter std test~~]		In [[category theory]], '''filtered categories''' generalize the notion of [[directed set]] understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered category). There is a dual notion of '''cofiltered''' category which will be recalled below.


			==Filtered categories==

			A [[Category (mathematics)\|category]] <math>J</math> is '''filtered''' when
			* it is not empty,
			* for every two objects <math>j</math> and <math>j'</math> in <math>J</math> there exists an object <math>k</math> and two arrows <math>f:j\to k</math> and <math>f':j'\to k</math> in <math>J</math>,
			* for every two parallel arrows <math>u,v:i\to j</math> in <math>J</math>, there exists an object <math>k</math> and an arrow <math>w:j\to k</math> such that <math>wu=wv</math>.

			A diagram is said to be of cardinality <math>\kappa</math> if the morphism set of its domain is of cardinality <math>\kappa</math>. A category <math>J</math> is filtered if and only if there is a cone over any finite diagram <math>d: D\to J</math>; more generally, for a regular cardinal <math>\kappa</math>, a category <math>J</math> is said to be <math>\kappa</math>-filtered if for every diagram <math>d</math> in <math>J</math> of cardinality smaller than <math>\kappa</math> there is a cone over <math>d</math>.

			A '''filtered colimit''' is a [[colimit]] of a [[functor]] <math>F:J\to C</math> where <math>J</math> is a filtered category. This readily generalizes to <math>\kappa</math>-filtered limits. An '''ind-object''' in a category <math>C</math> is a presheaf of sets <math>C^{op}\to Set</math> which is a small filtered colimit of representable presheaves. Ind-objects in a category <math>C</math> form a full subcategory <math>Ind(C)</math> in the category of functors <math>C^{op}\to Set</math>. The category <math>Pro(C)=Ind(C^{op})^{op}</math> of pro-objects in <math>C</math> is the opposite of the category of ind-objects in the opposite category <math>C^{op}</math>.

			==Cofiltered categories==
			A category <math>J</math> is cofiltered if the [[opposite category]] <math>J^{\mathrm{op}}</math> is filtered. In detail, a category is cofiltered when
			* it is not empty
			* for every two objects <math>j</math> and <math>j'</math> in <math>J</math> there exists an object <math>k</math> and two arrows <math>f:k\to j</math> and <math>f':k \to j'</math> in <math>J</math>,
			* for every two parallel arrows <math>u,v:j\to i</math> in <math>J</math>, there exists an object <math>k</math> and an arrow <math>w:k\to j</math> such that <math>uw=vw</math>.

			A '''cofiltered limit''' is a [[limit (category theory)\|limit]] of a [[functor]] <math>F:J \to C</math> where <math>J</math> is a cofiltered category.

			==References==
			* Artin, M., Grothendieck, A. and Verdier, J. L. ''Séminaire de Géométrie Algébrique du Bois Marie (SGA 4)''. Lecture Notes in Mathematics 269, Springer Verlag, 1972. Exposé I, 2.7.
			* {{Citation \| last1=Mac Lane \| first1=Saunders \| author1-link=Saunders Mac Lane \| title=[[Categories for the Working Mathematician]] \| publisher=[[Springer-Verlag]] \| location=Berlin, New York \| edition=2nd \| isbn=978-0-387-98403-2 \| year=1998}}, section IX.1.

			<references/>

			{{DEFAULTSORT:Filtered Category}}
			[[Category:Category theory]]

80.219.201.201: /* Refinement */

2012-05-18T20:06:03Z

Refinement

New page

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