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en>Sardanaphalus: /* External links */ template

2014-06-02T18:50:49Z

External links: template

← Older revision		Revision as of 20:50, 2 June 2014
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	~~[[image:FiveLemma.png\|350px\|thumb\|The commutative diagram used in the proof of the [[five lemma]].]]~~		Nothing to tell about me really.<br>Finally a part of this site.<br>I really hope I am useful in some way here.<br><br>Here is my webpage; [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities 2014]

	In mathematics, and especially in [[category theory]], a '''commutative diagram''' is a [[Diagram (category theory)\|diagram]] of objects (also known as ''vertices'') and [[morphism]]s (also known as ''arrows'' or ''edges'') such that all directed paths in the diagram with the same start and endpoints lead to ~~the same result by [[category (mathematics)#Definition\|composition]]~~. ~~Commutative diagrams play the role in category theory that [[equations]] play in [[algebra]] (see Barr-Wells, Section 1.7).~~

	~~Note that~~ a ~~diagram may not be commutative, i.e., the composition~~ of ~~different paths in the diagram may not give the same result. For clarification, phrases like "~~this ~~commutative diagram" or "the diagram commutes" may be used~~.

	~~==Examples==~~
	~~In the following diagram expressing the [[first isomorphism theorem]], commutativity means that~~ <~~math>f = \tilde{f} \circ \pi</math~~>:

	~~[[Image:First isomorphism theorem (plain).svg\|175px]]~~

	~~Below is a generic commutative square,~~ in ~~which <math>h \circ f = k \circ g</math>~~

	~~[[Image:Commutative square~~.~~svg\|150px]]~~

	~~===Symbols===~~
	In algebra texts, the type of [[morphism]] can be denoted with different arrow usages: [[monomorphism]]s with a <math>\hookrightarrow</math>, [[epimorphism]]s with a <math>\twoheadrightarrow</math>, and [[isomorphism]]s with a <~~math~~>~~\overset{\sim}{\rightarrow}~~<~~/math~~>. The dashed arrow typically represents the claim that the indicated morphism exists whenever the rest of the diagram holds. This is common enough that texts often do not explain the meanings of the different types of arrow.

	~~==Verifying commutativity==~~
	~~Commutativity makes sense for a [[polygon]] of any finite number of sides (including just 1 or 2), and a diagram~~ is ~~commutative if every polygonal subdiagram is commutative.~~

	~~==Diagram chasing==~~
	'''Diagram chasing''' is a method of [[mathematical proof]] used especially in [[homological algebra]]. Given a commutative diagram, a proof by diagram chasing involves the formal use of the properties of the diagram, such as [[injective]] or [[surjective]] maps, or [[exact sequence]]s. A [[syllogism]] is constructed, for which the graphical display of the diagram is just a visual aid. It follows that one ends up "chasing" elements around the diagram, until the desired element or result is constructed or verified.

	~~Examples of proofs by diagram chasing include those typically given for the [[five lemma]], the [[snake lemma]], the [[zig-zag lemma]], and the [[nine lemma]].~~

	~~== Diagrams as functors ==~~
	~~{{Main\|Diagram (category theory)}}~~

	~~A commutative diagram in a category ''C'' can be interpreted as a [[functor]] from an index category ''J'' to ''C~~;~~'' one calls the functor a '''~~[~~[diagram (category theory)\|diagram]].'''~~

	~~More formally, a commutative diagram is a visualization of a diagram indexed by a [[poset category]]:~~
	* one draws a node for every object in the index category,
	* an arrow for a generating set of morphisms,
	*:omitting identity maps and morphisms that can be expressed as compositions,
	* and the commutativity of the diagram (the equality of different compositions of maps between two objects) corresponds to the uniqueness of a map between two objects in a poset category.

	~~Conversely, given a commutative diagram, it defines a poset category:~~
	* the objects are the nodes,
	* there is a morphism between any two objects if and only if there is a (directed) path between the nodes,
	* with the relation that this morphism is unique (any composition of maps is defined by its domain and target: this is the commutativity axiom).

	However, not every diagram commutes (the notion of diagram strictly generalizes commutative diagram): 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 \rightrightarrows \bullet</math>, that is, <math>f,g\colon X \to Y</math>, sometimes called the [[free quiver]]), as used in the definition of [[equaliser (mathematics)\|equalizer]] need not commute. Further, diagrams may be messy or impossible to draw when the number of objects or morphisms is large (or even infinite).

	~~== See also ==~~
	* [[Mathematical diagram]]

	~~==References==~~
	*{{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).~~
	* {{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).

	~~== External links ==~~
	* [http://mathworld.wolfram.com/DiagramChasing.html Diagram Chasing] at [[MathWorld]]
	* [http://wildcatsformma.wordpress.com WildCats] is a category theory package for [[Mathematica]]. Manipulation and visualization of objects, [[morphism]]s, categories, [[functor]]s, [[natural transformation]]s.

	~~[[Category:Homological algebra]]~~
	~~[[Category:Category theory]]~~
	~~[[Category:Mathematical proofs]]~~
	~~[[Category:Mathematical terminology]]~~
	~~[[Category:Diagrams]~~]

en>D.Lazard: Reverted 1 edit by 68.6.92.19 (talk): For an inclusion, A = f(A). (TW)

2013-12-07T12:02:45Z

Reverted 1 edit by 68.6.92.19 (talk): For an inclusion, A = f(A). (TW)

← Older revision		Revision as of 14:02, 7 December 2013
Line 1:		Line 1:
	~~Some users~~ of ~~laptop or computer are aware which their computer become slower or have some mistakes after utilizing for~~ a ~~while. But most folks don~~'~~t learn how to speed up their computer plus some~~ of ~~them don~~'~~t dare~~ to ~~work it~~. ~~They always find some experts to keep~~ the ~~computer~~ in ~~good condition however they have to invest some funds on it~~. ~~Actually, we can do it by oneself~~. ~~There are numerous registry cleaner software~~ that ~~there are one of them online~~. ~~Some of them are free and we just have to download them~~. ~~After installing it~~, ~~this registry cleaner software~~ may ~~scan~~ the ~~registry~~. ~~If it found these errors~~, it may ~~report you plus you are able to delete them to keep your registry clean~~. ~~It is simple to operate and it really is~~ the ~~best way to repair registry.~~<br><br>~~If it~~ is ~~very not as big of~~ a ~~issue because we think it happens to be~~, ~~it will possibly~~ be ~~solved easily by running~~ a ~~Startup Repair or by System Restore Utility. Again it may be because effortless because running an anti-virus check or cleaning the registry.~~<br><br>~~Although this problem affects millions of computer users throughout the world~~, ~~there is an simple means to fix it. We see, there'~~s ~~1 reason for~~ a ~~slow loading computer~~, and ~~that'~~s ~~considering~~ a ~~PC cannot read~~ the ~~files it requires to run~~. ~~In a nutshell, this merely means~~ that ~~when you~~ do ~~anything on Windows~~, ~~it must read up on how to do it~~. It'~~s traditionally a especially~~ '~~dumb~~' ~~program, which has to have files to tell it to do everything.<br><br>If you feel you don~~'~~t have enough funds at the time to upgrade, then the greatest option~~ is ~~to free up several space by deleting~~ a ~~few~~ of ~~the unwelcome files and folders~~.~~<br><br>Besides~~, ~~should you can receive~~ a [~~http://bestregistrycleanerfix~~.~~com/system-mechanic iolo system mechanic~~] ~~that may work for we effectively and swiftly, then why not? There~~ is ~~one such system~~, ~~RegCure that~~ is ~~good plus complete~~. It ~~has attributes~~ that ~~different cleaners do not have. It~~ is ~~the many recommended registry cleaner today~~.~~<br><br>The most probable cause~~ of the ~~trouble is~~ the ~~program problem~~ - ~~Registry Errors! That is~~ the ~~reason why individuals who already have more than 2 G RAM on their computers are still consistently bothered by the issue~~.~~<br><br>The initially reason the computer~~ can be ~~slow is considering it needs more RAM~~. ~~You~~'~~ll notice this matter right away~~, ~~particularly should you have less than~~ a ~~gig~~ of ~~RAM~~. ~~Most new computers come with~~ a ~~least which much. While Microsoft states Windows XP may run on 128 MB~~, it and ~~Vista want at least~~ a ~~gig to run smoothly~~ and ~~let we to run multiple programs at once~~. ~~Fortunately~~, the ~~cost~~ of ~~RAM has dropped greatly~~, ~~plus you are able~~ to ~~get a gig installed for $100~~ or ~~less.~~<br><br>~~Ally Wood~~ is ~~a specialist software reviewer plus has worked inside CNET. Now she is working for her own review software business to provide feedback~~ to the ~~software creator plus has completed deep test~~ in ~~registry cleaner software~~. ~~After reviewing~~ the ~~most well known registry cleaner~~, ~~she has created complete review~~ on a ~~review site~~ for ~~you that may be accessed for free~~.		[[image:FiveLemma.png\|350px\|thumb\|The commutative diagram used in the proof of the [[five lemma]].]]

			In mathematics, and especially in [[category theory]], a '''commutative diagram''' is a [[Diagram (category theory)\|diagram]] of objects (also known as ''vertices'') and [[morphism]]s (also known as ''arrows'' or ''edges'') such that all directed paths in the diagram with the same start and endpoints lead to the same result by [[category (mathematics)#Definition\|composition]]. Commutative diagrams play the role in category theory that [[equations]] play in [[algebra]] (see Barr-Wells, Section 1.7).

			Note that a diagram may not be commutative, i.e., the composition of different paths in the diagram may not give the same result. For clarification, phrases like "this commutative diagram" or "the diagram commutes" may be used.

			==Examples==
			In the following diagram expressing the [[first isomorphism theorem]], commutativity means that <math>f = \tilde{f} \circ \pi</math>:

			[[Image:First isomorphism theorem (plain).svg\|175px]]

			Below is a generic commutative square, in which <math>h \circ f = k \circ g</math>

			[[Image:Commutative square.svg\|150px]]

			===Symbols===
			In algebra texts, the type of [[morphism]] can be denoted with different arrow usages: [[monomorphism]]s with a <math>\hookrightarrow</math>, [[epimorphism]]s with a <math>\twoheadrightarrow</math>, and [[isomorphism]]s with a <math>\overset{\sim}{\rightarrow}</math>. The dashed arrow typically represents the claim that the indicated morphism exists whenever the rest of the diagram holds. This is common enough that texts often do not explain the meanings of the different types of arrow.

			==Verifying commutativity==
			Commutativity makes sense for a [[polygon]] of any finite number of sides (including just 1 or 2), and a diagram is commutative if every polygonal subdiagram is commutative.

			==Diagram chasing==
			'''Diagram chasing''' is a method of [[mathematical proof]] used especially in [[homological algebra]]. Given a commutative diagram, a proof by diagram chasing involves the formal use of the properties of the diagram, such as [[injective]] or [[surjective]] maps, or [[exact sequence]]s. A [[syllogism]] is constructed, for which the graphical display of the diagram is just a visual aid. It follows that one ends up "chasing" elements around the diagram, until the desired element or result is constructed or verified.

			Examples of proofs by diagram chasing include those typically given for the [[five lemma]], the [[snake lemma]], the [[zig-zag lemma]], and the [[nine lemma]].

			== Diagrams as functors ==
			{{Main\|Diagram (category theory)}}

			A commutative diagram in a category ''C'' can be interpreted as a [[functor]] from an index category ''J'' to ''C;'' one calls the functor a '''[[diagram (category theory)\|diagram]].'''

			More formally, a commutative diagram is a visualization of a diagram indexed by a [[poset category]]:
			* one draws a node for every object in the index category,
			* an arrow for a generating set of morphisms,
			*:omitting identity maps and morphisms that can be expressed as compositions,
			* and the commutativity of the diagram (the equality of different compositions of maps between two objects) corresponds to the uniqueness of a map between two objects in a poset category.

			Conversely, given a commutative diagram, it defines a poset category:
			* the objects are the nodes,
			* there is a morphism between any two objects if and only if there is a (directed) path between the nodes,
			* with the relation that this morphism is unique (any composition of maps is defined by its domain and target: this is the commutativity axiom).

			However, not every diagram commutes (the notion of diagram strictly generalizes commutative diagram): 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 \rightrightarrows \bullet</math>, that is, <math>f,g\colon X \to Y</math>, sometimes called the [[free quiver]]), as used in the definition of [[equaliser (mathematics)\|equalizer]] need not commute. Further, diagrams may be messy or impossible to draw when the number of objects or morphisms is large (or even infinite).

			== See also ==
			* [[Mathematical diagram]]

			==References==
			*{{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).
			* {{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).

			== External links ==
			* [http://mathworld.wolfram.com/DiagramChasing.html Diagram Chasing] at [[MathWorld]]
			* [http://wildcatsformma.wordpress.com WildCats] is a category theory package for [[Mathematica]]. Manipulation and visualization of objects, [[morphism]]s, categories, [[functor]]s, [[natural transformation]]s.

			[[Category:Homological algebra]]
			[[Category:Category theory]]
			[[Category:Mathematical proofs]]
			[[Category:Mathematical terminology]]
			[[Category:Diagrams]]

193.48.126.34: /* Example */

2012-06-21T11:00:31Z

Example

New page

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