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en>Yobot: WP:CHECKWIKI error fixes using AWB (10687)

2015-01-05T09:11:48Z

WP:CHECKWIKI error fixes using AWB (10687)

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	~~In [[category theory]]~~, ~~a '''coequalizer''' (or '''coequaliser''') is a generalization of a [[quotient set\|quotient]] by an [[equivalence relation]] to objects in an arbitrary [[category (mathematics)\|category]]~~. ~~It is the categorical construction [[dual (category theory)\|dual]] to the [[equaliser (mathematics)\|equalizer]] (hence the name)~~.		Hello there, I am Adrianne and I totally really like that name. Vermont has always been simple home and I true love every day living suitable here. Gardening is what I do monthly. I am a people manager but rather soon I'll be on my own. You can find my world wide web here: http://circuspartypanama.com<br><br>Take a look at my webpage ... clash of clans hack cydia ([http://circuspartypanama.com circuspartypanama.com])

	~~== Definition ==~~

	~~A '''coequalizer'''~~ is ~~a [[colimit]] of the diagram consisting of two objects ''X'' and ''Y'' and two parallel [[morphism]]s ''f'', ''g'' : ''X'' → ''Y''~~.

	~~More explicitly,~~ a ~~coequalizer can be defined as an object ''Q'' together with a morphism ''q'' : ''Y'' → ''Q'' such that ''q'' ∘ ''f'' = ''q'' ∘ '~~'~~g''. Moreover, the pair (''Q'', ''q'') must~~ be [[universal property\|universal]] in the sense that given any other such pair (''Q''′, ''q''′) there exists a unique morphism ''u'' : ''Q'' → ''Q''′ for which the following diagram [[commutative diagram\|commutes]]:

	~~<div style="text-align: center;">[[Image:Coequalizer-01.png]]</div>~~

	As with all [[universal construction]]s, a coequalizer, if it exists, is unique [[up to]] a unique [[isomorphism]] (this is why, by abuse of language, one sometimes speaks of "the" coequalizer of two parallel arrows).

	It can ~~be shown that a coequalizer ''q'' is an [[epimorphism]] in any category.~~

	~~== Examples ==~~

	*In the [[category of sets]], the coequalizer of two [[function (mathematics)\|function]]s ''f'', ''g'' : ''X'' → ''Y'' is the [[quotient set\|quotient]] of ''Y'' by the smallest [[equivalence relation]] <math>~\sim</math> such that for every <math>x\in X</math>, we have <math>f(x)\sim g(x)</math>.<ref>{{cite book \|last1=Barr \|first1=Michael \|authorlink1=Michael Barr (mathematician) \|last2=Wells \|first2=Charles \|authorlink2=Charles Wells (mathematician) \|year=1998 \|title=Category theory for computing science \|page=278 \|url=http://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf \|accessdate=2013-07-25 \|format=PDF}}</ref> In particular, if ''R'' is an equivalence relation on a set ''Y'', and ''r''<sub>1</sub>, ''r''<sub>2</sub> are the natural projections (''R'' ⊂ ''Y'' × ''Y'') → ''Y'' then the coequalizer of ''r''<sub>1</sub> and ''r''<sub>2</sub> is the quotient set ''Y''/''R''.

	*The coequalizer in the [[category of groups]] is very similar. Here if ''f'', ''g'' : ''X'' → ''Y'' are [[group homomorphism]]s, their coequalizer is the [[quotient group\|quotient]] of ''Y'' by the [[Normal closure (group theory)\|normal closure]] of the set
	~~:<math>S=\{f(x)g(x)^{-1}\ \|\ x\in X\}</math>~~

	*For [[abelian group]]s the coequalizer is particularly simple. It is just the [[factor group]] ''Y'' / im(''f'' – ''g''). (This is the [[cokernel]] of the morphism ''f'' – ''g''; see the next section).

	*In the [[category of topological spaces]], the circle object <~~math~~>~~S^1~~<~~/math~~> ~~can be viewed as the coequalizer of the two inclusion maps from the standard 0-simplex to the standard 1-simplex.~~

	*Coequalisers can be large: There are exactly two functors from the category '''1''' having one object and one identity arrow, to the category '''2''' with two objects and exactly one non-identity arrow going between them. The coequaliser of these two functors is the monoid of natural numbers under addition, considered as a one-object category. In particular, this shows that while every coequalising arrow is [[Epimorphism\|epic]], it is not necessarily [[Surjective_function\|surjective]].

	~~== Properties ==~~
	*In a [[topos]], every [[epimorphism]] is the coequalizer of its kernel pair.
	*Every coequalizer is an epimorphism.

	~~== Special cases ==~~

	~~In categories with [[zero morphism]]s, one can define a ''[[cokernel]]'' of~~ a ~~morphism ''f'' as the coequalizer of ''f'' and the parallel zero morphism.~~

	~~In [[preadditive category\|preadditive categories]] it makes sense to add and subtract morphisms (the [[hom-set]]s actually form [[abelian group]]s)~~. ~~In such categories, one can define the coequalizer of two morphisms ''f'' and ''g'' as the cokernel of their difference:~~
	~~:coeq(''f'', ''g'') = coker(''g'' – ''f'').~~

	~~A stronger notion is that of an '''absolute coequalizer''', this is a coequalizer that is preserved under all [[functors]]~~.
	Formally, an absolute coequalizer of a pair <math>f,g: X \to Y</math> in a category C is a coequalizer as defined above but with the added property that given any functor <math>F: C \to D</math> F(Q) together with F(q) is the coequalizer of F(f) and F(g) in the category D. ~~[[Split coequalizers]] are examples~~ of ~~absolute coequalizers.~~

	~~== See also ==~~
	*[[equalizer (~~mathematics)]]~~
	*[[coproduct]]
	*[[Pushout (category theory)\|pushout]]

	~~==Notes==~~
	~~{{reflist}}~~

	~~== References ==~~
	*[[Saunders Mac Lane]]: [[Categories for the Working Mathematician]], Second Edition, 1998.
	*Coequalizers - page 65
	*Absolute coequalizers - page 149

	~~== External links ==~~
	*[http://~~www~~.~~j-paine~~.~~org/cgi-bin/webcats/webcats.php Interactive Web page~~ ] ~~which generates examples of coequalizers in the category of finite sets. Written by [http://www.j-paine.org/ Jocelyn Paine].~~

	~~[[Category:Limits (category theory~~)]]

en>BD2412: /* Usage example */Fixing links to disambiguation pages, improving links, other minor cleanup tasks using AWB

2013-12-27T02:23:44Z

Usage example: Fixing links to disambiguation pages, improving links, other minor cleanup tasks using AWB

← Older revision		Revision as of 04:23, 27 December 2013
Line 1:		Line 1:
	~~Beforehand playing~~ a ~~new video game, read the gain~~ a ~~advantage book. Most online have~~ a ~~book you'll can purchase separately~~. ~~You may want when you need~~ to ~~consider doing this but reading it before an individual play, or even despite~~ the ~~fact you are playing. This way, you can possibly get~~ the ~~most in of your game play~~.~~<br><br>~~		In [[category theory]], a '''coequalizer''' (or '''coequaliser''') is a generalization of a [[quotient set\|quotient]] by an [[equivalence relation]] to objects in an arbitrary [[category (mathematics)\|category]]. It is the categorical construction [[dual (category theory)\|dual]] to the [[equaliser (mathematics)\|equalizer]] (hence the name).

	~~Video gaming are fun to practice with your kids. If you have any queries regarding wherever~~ and ~~how to use~~ [~~http~~:~~//prometeu~~.~~net code triche clash of clans]~~, ~~you~~ can ~~call us at our own web page~~. ~~This can help you learn much more about your kid~~'~~s interests~~. ~~Sharing interests~~ with ~~your kids like~~ this ~~can conjointly create great conversations~~. It ~~also gives you~~ an ~~opportunity to monitor growth and development~~ of ~~their skills.~~<br><br>~~Prevent purchasing big~~ title betting games near their launch dating. Waiting means that you're prone to achieve clash of clans cheats after having a patch or two is carrying emerge to mend manifest holes and bugs may possibly impact your pleasure along with [http://~~Dict~~.~~Leo~~.~~org~~/~~?search~~=~~game+play game play]. Near keep~~ an ~~eye out for titles from galleries which~~ are ~~understood for good patching~~ and ~~support.~~<br><br>~~Hold note~~ of ~~how a great money your teen~~ is ~~shelling out for gambling~~. ~~These kinds towards products aren~~'~~t cheap but then there~~ is ~~highly~~ the ~~option~~ of ~~purchasing much more add~~-~~ons near~~ the ~~game itself~~. ~~Establish month-to-month and to choose from restrictions on~~ the ~~size~~ of ~~money that has~~ the ~~ability to be spent on video~~. ~~Also~~, ~~have conversations that has~~ the ~~youngsters about financing.~~<br><br>~~Actual not only provides complete tools, there is also Clash~~ of ~~Clans identify no survey by anyone. Strict anti ban system make users~~ to ~~utilize shed pounds and play without type of hindrance~~. ~~If enthusiastic gamers~~ are ~~interested in obtaining~~ the ~~program~~, ~~they are absolutely required~~ to ~~visit fantastic site~~ and ~~obtain each of our hack tool trainer immediately~~. The ~~name~~ of the ~~website is Amazing Cheats. A number~~ of ~~websites have different types out of software by which people can get past tough stages in the game~~.~~<br><br>An individual are playing a applying game~~, ~~and you haven't got any experience with~~ it, ~~set~~ the ~~difficulty even to rookie~~. ~~This~~ is ~~considered help you pick in on~~ the ~~unique has~~ of the ~~game learn your way almost~~ the ~~field~~. In the ~~case when you set it superior than~~ that, ~~you are likely~~ to ~~get frustrated and so not have~~ any ~~good.~~<br><br>~~In which to master game play to shooter video games, pro your weapons. Discover everything there~~ is ~~understand about each~~ and ~~each single weapon style~~ in ~~sport~~. ~~Each weapon excels while in certain ways~~, ~~but falls short~~ in ~~others~~. ~~When you know that pluses and minuses relating to each weapon, you does use them to overly full advantage~~.		== Definition ==

			A '''coequalizer''' is a [[colimit]] of the diagram consisting of two objects ''X'' and ''Y'' and two parallel [[morphism]]s ''f'', ''g'' : ''X'' → ''Y''.

			More explicitly, a coequalizer can be defined as an object ''Q'' together with a morphism ''q'' : ''Y'' → ''Q'' such that ''q'' ∘ ''f'' = ''q'' ∘ ''g''. Moreover, the pair (''Q'', ''q'') must be [[universal property\|universal]] in the sense that given any other such pair (''Q''′, ''q''′) there exists a unique morphism ''u'' : ''Q'' → ''Q''′ for which the following diagram [[commutative diagram\|commutes]]:

			<div style="text-align: center;">[[Image:Coequalizer-01.png]]</div>

			As with all [[universal construction]]s, a coequalizer, if it exists, is unique [[up to]] a unique [[isomorphism]] (this is why, by abuse of language, one sometimes speaks of "the" coequalizer of two parallel arrows).

			It can be shown that a coequalizer ''q'' is an [[epimorphism]] in any category.

			== Examples ==

			*In the [[category of sets]], the coequalizer of two [[function (mathematics)\|function]]s ''f'', ''g'' : ''X'' → ''Y'' is the [[quotient set\|quotient]] of ''Y'' by the smallest [[equivalence relation]] <math>~\sim</math> such that for every <math>x\in X</math>, we have <math>f(x)\sim g(x)</math>.<ref>{{cite book \|last1=Barr \|first1=Michael \|authorlink1=Michael Barr (mathematician) \|last2=Wells \|first2=Charles \|authorlink2=Charles Wells (mathematician) \|year=1998 \|title=Category theory for computing science \|page=278 \|url=http://www.math.mcgill.ca/triples/Barr-Wells-ctcs.pdf \|accessdate=2013-07-25 \|format=PDF}}</ref> In particular, if ''R'' is an equivalence relation on a set ''Y'', and ''r''<sub>1</sub>, ''r''<sub>2</sub> are the natural projections (''R'' ⊂ ''Y'' × ''Y'') → ''Y'' then the coequalizer of ''r''<sub>1</sub> and ''r''<sub>2</sub> is the quotient set ''Y''/''R''.

			*The coequalizer in the [[category of groups]] is very similar. Here if ''f'', ''g'' : ''X'' → ''Y'' are [[group homomorphism]]s, their coequalizer is the [[quotient group\|quotient]] of ''Y'' by the [[Normal closure (group theory)\|normal closure]] of the set
			:<math>S=\{f(x)g(x)^{-1}\ \|\ x\in X\}</math>

			*For [[abelian group]]s the coequalizer is particularly simple. It is just the [[factor group]] ''Y'' / im(''f'' – ''g''). (This is the [[cokernel]] of the morphism ''f'' – ''g''; see the next section).

			*In the [[category of topological spaces]], the circle object <math>S^1</math> can be viewed as the coequalizer of the two inclusion maps from the standard 0-simplex to the standard 1-simplex.

			*Coequalisers can be large: There are exactly two functors from the category '''1''' having one object and one identity arrow, to the category '''2''' with two objects and exactly one non-identity arrow going between them. The coequaliser of these two functors is the monoid of natural numbers under addition, considered as a one-object category. In particular, this shows that while every coequalising arrow is [[Epimorphism\|epic]], it is not necessarily [[Surjective_function\|surjective]].

			== Properties ==
			*In a [[topos]], every [[epimorphism]] is the coequalizer of its kernel pair.
			*Every coequalizer is an epimorphism.

			== Special cases ==

			In categories with [[zero morphism]]s, one can define a ''[[cokernel]]'' of a morphism ''f'' as the coequalizer of ''f'' and the parallel zero morphism.

			In [[preadditive category\|preadditive categories]] it makes sense to add and subtract morphisms (the [[hom-set]]s actually form [[abelian group]]s). In such categories, one can define the coequalizer of two morphisms ''f'' and ''g'' as the cokernel of their difference:
			:coeq(''f'', ''g'') = coker(''g'' – ''f'').

			A stronger notion is that of an '''absolute coequalizer''', this is a coequalizer that is preserved under all [[functors]].
			Formally, an absolute coequalizer of a pair <math>f,g: X \to Y</math> in a category C is a coequalizer as defined above but with the added property that given any functor <math>F: C \to D</math> F(Q) together with F(q) is the coequalizer of F(f) and F(g) in the category D. [[Split coequalizers]] are examples of absolute coequalizers.

			== See also ==
			*[[equalizer (mathematics)]]
			*[[coproduct]]
			*[[Pushout (category theory)\|pushout]]

			==Notes==
			{{reflist}}

			== References ==
			*[[Saunders Mac Lane]]: [[Categories for the Working Mathematician]], Second Edition, 1998.
			*Coequalizers - page 65
			*Absolute coequalizers - page 149

			== External links ==
			*[http://www.j-paine.org/cgi-bin/webcats/webcats.php Interactive Web page ] which generates examples of coequalizers in the category of finite sets. Written by [http://www.j-paine.org/ Jocelyn Paine].

			[[Category:Limits (category theory)]]

en>ZéroBot: r2.7.1) (Robot: Adding ja:積分変換

2012-07-09T13:59:25Z

r2.7.1) (Robot: Adding ja:積分変換

New page

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