Generalized coordinates - Revision history

en>Maschen: /* Constraint equations */ replace my incorrect image with better ones

2014-12-06T23:39:46Z

Constraint equations: replace my incorrect image with better ones

← Older revision		Revision as of 01:39, 7 December 2014
Line 1:		Line 1:
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en>Renatops: /* Virtual work */ fixed missing L subscripts in generalized forces

2014-02-26T15:29:55Z

Virtual work: fixed missing L subscripts in generalized forces

← Older revision		Revision as of 17:29, 26 February 2014
Line 1:		Line 1:
	~~In [[mathematics]], especially in the field of [[category theory]], the concept of '''injective object''~~' is a generalization of the concept of [[injective module]]. This concept is important in [[homotopy theory]] and in theory of [[model category\|model categories]]. The dual notion is that of a [[projective object]].		I'm a 47 years old and study at the university (Philosophy).<br>In my free time I teach myself Arabic. I have been there and look forward to returning sometime in the future. I like to read, preferably on my kindle. I really love to watch The Simpsons and Breaking Bad as well as documentaries about nature. I enjoy Leaf collecting and pressing.<br><br>Here is my page: [https://www.youtube.com/watch?v=eJEoN68iK0k lien]

	~~==General Definition==~~

	~~Let <math>\mathfrak{C}</math> be a category and let <math>\mathcal{H}</math> be~~ a ~~class of morphisms of <math>\mathfrak{C}</math>.~~

	~~An object <math>Q</math> of <math>\mathfrak{C}</math> is said to be '''''<math>\mathcal{H}</math>''-injective''' if for every morphism <math>f: A \to Q</math>~~ and ~~every morphism <math>h: A \to B</math> in <math>\mathcal{H}</math> there exists a morphism <math>g: B \to Q</math> extending (~~the ~~domain of) <math>f</math>, i.e <math> gh = f</math>. In other words, <math>Q</math> is injective iff any <math>\mathcal{H}</math>-morphism into <math>Q</math> extends~~ (~~via composition on the left~~) ~~to a morphism into <math>Q</math>~~.

	~~The morphism <math>g</math> in the above definition is not required to be uniquely determined by <math>h</math> and~~ <~~math~~>~~f</math>.~~

	In ~~a locally small category, it is equivalent to require that the [[hom functor]] <math>Hom_{\mathfrak{C}}(-,Q)</math> carries <math>\mathcal{H}</math>-morphisms to epimorphisms (surjections)~~.

	~~The classical choice for <math>\mathcal{H}</math> is the class of [[monomorphism]]s, in this case, the expression '''injective object''' is used.~~

	~~==Abelian case==~~

	If <math>\mathfrak{C}</math> is an [[abelian category]], an object ''A'' of <math>\mathfrak{C}</math> is injective iff its [[hom functor]] Hom<sub>'''C'''</sub>(–,''A'') is [[exact functor\|exact]].

	~~The abelian case was the original framework for the notion of injectivity.~~

	~~==Enough injectives==~~

	~~Let <math>\mathfrak{C}</math> be a category, ''H'' a class of morphisms of <math>\mathfrak{C}</math> ; the category <math>\mathfrak{C}</math> is said to ''~~have ~~enough H-injectives'' if for every object ''X'' of <math>\mathfrak{C}</math>,~~ there ~~exist a ''H''-morphism from ''X''~~ to ~~an ''H''-injective object.~~

	~~==Injective hull==~~

	~~A ''H''-morphism ''g''~~ in ~~<math>\mathfrak{C}</math> is called '''''H''-essential''' if for any morphism ''f'',~~ the ~~composite ''fg'' is in ''H'' only if ''f'' is in ''H''~~.

	~~If ''f'' is a ''H''-essential ''H''-morphism with a domain ''X'' and an ''H''-injective codomain ''G''~~, ~~''G'' is called an '''''H''-injective hull''' of ''X''~~. ~~This ''H''-injective hull is then unique up~~ to ~~a canonical isomorphism.~~

	~~==Examples==~~

	*In the category of [[Abelian group]]s and ~~[[group homomorphism]]s, an injective object is a [[divisible group]].~~
	*In the category of [[Module (mathematics)\|modules]] and [[module homomorphism]]s, ''R''-Mod, an injective object is an [[injective module]]. ''R''-Mod has [[injective hull]]s (as ~~a consequence, R-Mod has enough injectives).~~
	*In the category of [[metric space]]s and [[nonexpansive mapping]]s, [[Category of metric spaces\|Met]], an injective object is an [[injective metric space]], and the injective hull of a metric space is its [[tight span]].
	*In the category of [[T0 space]]s and ~~[[continuous mapping]]s, an injective object is always a [[Scott topology]] on a [[continuous lattice]] therefore it is always [[Sober space\|sober]] and [[locally compact]]~~.
	*In the category of [[simplicial set]]s, the injective objects with respect to the class of anodyne extensions are [[Kan complex]]es.
	*In the category of partially ordered sets and monotonic functions between posets, the [[complete lattice]]s form the injective objects for [[order-embedding]]s, and the [[Dedekind–MacNeille completion]] of a partially ordered set is its injective hull.
	*One also talks about injective objects in more general categories, for instance in [[functor category\|functor categories]] or in categories of [[sheaf (mathematics)\|sheaves]] of O<~~sub~~>~~''X''~~<~~/sub~~> ~~modules over some~~ [~~[ringed space]] (''X'',O<sub>''X''<~~/~~sub>).~~

	~~==References==~~
	*J. Rosicky, Injectivity and accessible categories
	*F. ~~Cagliari and S~~. ~~Montovani, T<sub>0<~~/~~sub>-reflection and injective hulls of fibre spaces~~



	~~[[Category:Category theory]]~~

	~~[[de:Injektiver Modul#Injektive Moduln]~~]

50.158.36.58: Removing unnecessary information

2013-10-31T02:27:42Z

Removing unnecessary information

← Older revision		Revision as of 04:27, 31 October 2013
Line 1:		Line 1:
	~~Adrianne~~ is ~~what you~~ is ~~able~~ to ~~call me but I don~~'~~t like when guys~~ and ~~women use my full name~~. ~~What I love doing~~ is ~~always~~ to ~~play [http:~~//~~Croquet~~.~~org~~/ ~~croquet] now I have your time~~ to ~~take on issues~~. The ~~job I~~'~~ve been occupying for many years~~ is an ~~portion clerk. My husband and~~ [~~http:~~//~~www~~.~~adobe~~.~~com~~/~~cfusion~~/~~search~~/~~index~~.~~cfm?term~~=~~&I+decide&loc~~=~~en_us&siteSection~~=~~home I decide] to reside~~ in ~~Guam but I will own~~ to ~~move in per year or two~~. ~~You~~ has the ~~capability~~ to ~~find my website here: http:~~//~~prometeu~~.~~net~~<br><br>~~Have a look at my web site; how to hack clash~~ of ~~clans (~~[~~http~~:~~//prometeu.net just click the following web site~~])		In [[mathematics]], especially in the field of [[category theory]], the concept of '''injective object''' is a generalization of the concept of [[injective module]]. This concept is important in [[homotopy theory]] and in theory of [[model category\|model categories]]. The dual notion is that of a [[projective object]].

			==General Definition==

			Let <math>\mathfrak{C}</math> be a category and let <math>\mathcal{H}</math> be a class of morphisms of <math>\mathfrak{C}</math>.

			An object <math>Q</math> of <math>\mathfrak{C}</math> is said to be '''''<math>\mathcal{H}</math>''-injective''' if for every morphism <math>f: A \to Q</math> and every morphism <math>h: A \to B</math> in <math>\mathcal{H}</math> there exists a morphism <math>g: B \to Q</math> extending (the domain of) <math>f</math>, i.e <math> gh = f</math>. In other words, <math>Q</math> is injective iff any <math>\mathcal{H}</math>-morphism into <math>Q</math> extends (via composition on the left) to a morphism into <math>Q</math>.

			The morphism <math>g</math> in the above definition is not required to be uniquely determined by <math>h</math> and <math>f</math>.

			In a locally small category, it is equivalent to require that the [[hom functor]] <math>Hom_{\mathfrak{C}}(-,Q)</math> carries <math>\mathcal{H}</math>-morphisms to epimorphisms (surjections).

			The classical choice for <math>\mathcal{H}</math> is the class of [[monomorphism]]s, in this case, the expression '''injective object''' is used.

			==Abelian case==

			If <math>\mathfrak{C}</math> is an [[abelian category]], an object ''A'' of <math>\mathfrak{C}</math> is injective iff its [[hom functor]] Hom<sub>'''C'''</sub>(–,''A'') is [[exact functor\|exact]].

			The abelian case was the original framework for the notion of injectivity.

			==Enough injectives==

			Let <math>\mathfrak{C}</math> be a category, ''H'' a class of morphisms of <math>\mathfrak{C}</math> ; the category <math>\mathfrak{C}</math> is said to ''have enough H-injectives'' if for every object ''X'' of <math>\mathfrak{C}</math>, there exist a ''H''-morphism from ''X'' to an ''H''-injective object.

			==Injective hull==

			A ''H''-morphism ''g'' in <math>\mathfrak{C}</math> is called '''''H''-essential''' if for any morphism ''f'', the composite ''fg'' is in ''H'' only if ''f'' is in ''H''.

			If ''f'' is a ''H''-essential ''H''-morphism with a domain ''X'' and an ''H''-injective codomain ''G'', ''G'' is called an '''''H''-injective hull''' of ''X''. This ''H''-injective hull is then unique up to a canonical isomorphism.

			==Examples==

			*In the category of [[Abelian group]]s and [[group homomorphism]]s, an injective object is a [[divisible group]].
			*In the category of [[Module (mathematics)\|modules]] and [[module homomorphism]]s, ''R''-Mod, an injective object is an [[injective module]]. ''R''-Mod has [[injective hull]]s (as a consequence, R-Mod has enough injectives).
			*In the category of [[metric space]]s and [[nonexpansive mapping]]s, [[Category of metric spaces\|Met]], an injective object is an [[injective metric space]], and the injective hull of a metric space is its [[tight span]].
			*In the category of [[T0 space]]s and [[continuous mapping]]s, an injective object is always a [[Scott topology]] on a [[continuous lattice]] therefore it is always [[Sober space\|sober]] and [[locally compact]].
			*In the category of [[simplicial set]]s, the injective objects with respect to the class of anodyne extensions are [[Kan complex]]es.
			*In the category of partially ordered sets and monotonic functions between posets, the [[complete lattice]]s form the injective objects for [[order-embedding]]s, and the [[Dedekind–MacNeille completion]] of a partially ordered set is its injective hull.
			*One also talks about injective objects in more general categories, for instance in [[functor category\|functor categories]] or in categories of [[sheaf (mathematics)\|sheaves]] of O<sub>''X''</sub> modules over some [[ringed space]] (''X'',O<sub>''X''</sub>).

			==References==
			*J. Rosicky, Injectivity and accessible categories
			*F. Cagliari and S. Montovani, T<sub>0</sub>-reflection and injective hulls of fibre spaces



			[[Category:Category theory]]

			[[de:Injektiver Modul#Injektive Moduln]]

en>Zorrobot: r2.7.3) (Robot: Adding sv:Generaliserade koordinater

2012-07-18T15:41:59Z

r2.7.3) (Robot: Adding sv:Generaliserade koordinater

New page

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