98.230.20.101: rm vandalism

2014-12-16T18:48:09Z

rm vandalism

en>Monkbot: /* Misconceptions about recoil */Fix CS1 deprecated date parameter errors

2014-01-29T04:23:04Z

Misconceptions about recoil: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 06:23, 29 January 2014
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	~~Individual who wrote~~ the is ~~called Eusebio~~. ~~South Carolina~~ is ~~his birth place~~. The ~~best-loved hobby for him as well~~ as the ~~his kids~~ is to ~~fish and he~~'s ~~resulted~~ in being ~~doing~~ it ~~for~~ a ~~long time~~. ~~Filing has been his profession for~~ a ~~short time~~. Go to ~~his website to discover~~ a ~~out more: http:~~//~~prometeu~~.~~net~~<br><br>~~Feel free~~ to ~~surf~~ to ~~my website;~~ [~~http~~://~~prometeu~~.~~net Clash Of Clans Hack Tool~~]		In [[mathematics]], a '''direct limit''' (also called '''inductive limit''') is a [[limit (category theory)\|colimit]] of a "directed family of objects". We will first give the definition for [[algebraic structure]]s like [[group (mathematics)\|groups]] and [[module (mathematics)\|modules]], and then the general definition which can be used in any [[category (mathematics)\|category]].

			==Formal definition==

			=== Algebraic objects ===

			In this section objects are understood to be [[Set (mathematics)\|sets]] with a given [[algebraic structure]] such as [[group (mathematics)\|groups]], [[ring (mathematics)\|rings]], [[module (mathematics)\|modules]] (over a fixed ring), [[algebra over a field\|algebras]] (over a fixed field), etc. With this in mind, ''[[homomorphism]]s'' are understood in the corresponding setting ([[group homomorphism]]s, etc.).

			Start with the definition of a '''direct system''' of objects and homomorphisms. Let <math>\langle I,\le\rangle</math> be a [[directed set]]. Let <math>\{A_i : i\in I\}</math> be a family of objects [[index set\|indexed]] by <math>I\,</math> and <math> f_{ij}: A_i \rightarrow A_j </math> be a homomorphism for all <math>i \le j</math> with the following properties:
			# <math>f_{ii}\,</math> is the identity of <math>A_i\,</math>, and
			# <math>f_{ik}= f_{jk}\circ f_{ij}</math> for all <math>i\le j\le k</math>.
			Then the pair <math>\langle A_i,f_{ij}\rangle</math> is called a direct system over <math>I\,</math>.

			The underlying set of the '''direct limit''', <math>A\,</math>, of the direct system <math>\langle A_i,f_{ij}\rangle</math> is defined as the [[disjoint union]] of the <math>A_i\,</math>'s [[Modulo (jargon)\|modulo]] a certain [[equivalence relation]] <math>\sim\,</math>:
			:<math>\varinjlim A_i = \bigsqcup_i A_i\bigg/\sim.</math>
			Here, if <math>x_i\in A_i</math> and <math>x_j\in A_j</math>, <math>x_i\sim\, x_j</math> if there is some <math>k\in I</math> such that <math>f_{ik}(x_i) = f_{jk}(x_j)\,</math>.
			Heuristically, two elements in the disjoint union are equivalent [[if and only if]] they "eventually become equal" in the direct system. An equivalent formulation that highlights the duality to the [[inverse limit]] is that an element is equivalent to all its images under the maps of the directed system, i.e. <math>x_i\sim\, f_{ik}(x_i)</math>.

			One naturally obtains from this definition ''canonical [[morphism]]s'' <math>\phi_i: A_i\rightarrow A</math> sending each element to its equivalence class. The algebraic operations on <math>A\,</math> are defined via these maps in the obvious manner.

			An important property is that taking direct limits in the category of [[module (mathematics)\|modules]] is an [[exact functor]].

			===Direct limit over a direct system in a category===
			The direct limit can be defined in an arbitrary [[category (mathematics)\|category]] <math>\mathcal{C}</math> by means of a [[universal property]]. Let <math>\langle X_i, f_{ij}\rangle</math> be a direct system of objects and morphisms in <math>\mathcal{C}</math> (same definition as above). The direct limit of this system is an object <math>X\,</math> in <math>\mathcal{C}</math> together with morphisms <math>\phi_i: X_i\rightarrow X</math> satisfying <math>\phi_i =\phi_j \circ f_{ij}</math>. The pair <math>\langle X, \phi_i\rangle</math> must be universal in the sense that for any other such pair <math>\langle Y, \psi_i\rangle</math> there exists a unique morphism <math> u:X\rightarrow Y</math> making the diagram
			<div style="text-align: center;">[[Image:DirectLimit-01.png]]</div>

			[[commutative diagram\|commute]] for all ''i'', ''j''. The direct limit is often denoted
			:<math>X = \varinjlim X_i</math>
			with the direct system <math>\langle X_i, f_{ij}\rangle</math> being understood.

			Unlike for algebraic objects, the direct limit may not exist in an arbitrary category. If it does, however, it is unique in a strong sense: given another direct limit ''X''′ there exists a ''unique'' [[isomorphism]] ''X''′ → ''X'' commuting with the canonical morphisms.

			We note that a direct system in a category <math>\mathcal{C}</math> admits an alternative description in terms of [[functor]]s. Any directed poset <math>\langle I,\le \rangle</math> can be considered as a [[small category]] <math>\mathcal{I}</math> where the morphisms consist of arrows <math>i\rightarrow j</math> [[if and only if]] <math>i\le j</math>. A direct system is then just a [[covariant functor]] <math>\mathcal{I}\rightarrow \mathcal{C}</math>. In this case a direct limit is a universal (initial) [[Cone (category theory)\|co-cone]].

			==Examples==
			*A collection of subsets <math>M_i</math> of a set ''M'' can be [[Partial order\|partially ordered]] by inclusion. If the collection is directed, its direct limit is the union <math>\bigcup M_i</math>.
			*Let ''I'' be any directed set with a [[greatest element]] ''m''. The direct limit of any corresponding direct system is isomorphic to ''X''<sub>''m''</sub> and the canonical morphism φ<sub>''m''</sub>: ''X''<sub>''m''</sub> → ''X'' is an isomorphism.
			*Let ''p'' be a [[prime number]]. Consider the direct system composed of the groups '''Z'''/''p''<sup>''n''</sup>'''Z''' and the homomorphisms '''Z'''/''p''<sup>''n''</sup>'''Z''' → '''Z'''/''p''<sup>''n''+1</sup>'''Z''' which are induced by multiplication by ''p''. The direct limit of this system consists of all the [[roots of unity]] of order some power of ''p'', and is called the [[Prüfer group]] '''Z'''(''p''<sup>∞</sup>).
			*Let ''F'' be a ''C''-valued [[sheaf (mathematics)\|sheaf]] on a [[topological space]] ''X''. Fix a point ''x'' in ''X''. The open neighborhoods of ''x'' form a directed poset ordered by inclusion (''U'' ≤ ''V'' if and only if ''U'' contains ''V''). The corresponding direct system is (''F''(''U''), ''r''<sub>''U'',''V''</sub>) where ''r'' is the restriction map. The direct limit of this system is called the ''[[stalk (mathematics)\|stalk]]'' of ''F'' at ''x'', denoted ''F''<sub>''x''</sub>. For each neighborhood ''U'' of ''x'', the canonical morphism ''F''(''U'') → ''F''<sub>''x''</sub> associates to a section ''s'' of ''F'' over ''U'' an element ''s''<sub>''x''</sub> of the stalk ''F''<sub>''x''</sub> called the ''[[germ (mathematics)\|germ]]'' of ''s'' at ''x''.
			*Direct limits in the [[category of topological spaces]] are given by placing the [[final topology]] on the underlying set-theoretic direct limit.
			*Direct limits are linked to [[inverse limit]]s via
			:<math>\mathrm{Hom} (\varinjlim X_i, Y) = \varprojlim \mathrm{Hom} (X_i, Y).</math>
			Consider a sequence {''A<sub>n</sub>'', ''φ<sub>n</sub>''} where ''A<sub>n</sub>'' is a [[C-algebra]] and ''φ<sub>n</sub>'' : ''A<sub>n</sub>'' → ''A''<sub>''n'' + 1</sub> is a -homomorphism. The C-analog of the direct limit construction gives a C*-algebra satisfying the universal property above.

			==Related constructions and generalizations==
			The [[Dual (category theory)\|categorical dual]] of the direct limit is called the [[inverse limit]] (or projective limit). More general concepts are the [[limit (category theory)\|limits and colimits]] of category theory. The terminology is somewhat confusing: direct limits are colimits while inverse limits are limits.

			==See also==
			* [[Inverse limit\|Inverse, or projective limit]]

			==References==
			* {{Citation \| last1=Bourbaki \| first1=Nicolas \| author1-link=Nicolas Bourbaki \| title=Elements of mathematics. Theory of sets \| publisher=Hermann \| location=Paris \| series=Translated from the French \| mr=0237342 \| year=1968}}.
			* {{Citation \|authorlink=Saunders Mac Lane \|last=Mac Lane \|first=Saunders \|year=1998 \|title=[[Categories for the Working Mathematician]] \|edition=2nd \|series=Graduate Texts in Mathematics \|volume=5 \|publisher=Springer-Verlag}}.

			{{DEFAULTSORT:Direct Limit}}
			[[Category:Limits (category theory)]]
			[[Category:Abstract algebra]]

en>ZéroBot: r2.7.1) (Robot: Adding ca:Retrocés

2012-06-14T23:16:54Z

r2.7.1) (Robot: Adding ca:Retrocés

New page

Individual who wrote the is called Eusebio. South Carolina is his birth place. The best-loved hobby for him as well as the his kids is to fish and he's resulted in being doing it for a long time. Filing has been his profession for a short time. Go to his website to discover a out more: http://prometeu.net<br><br>Feel free to surf to my website; [http://prometeu.net Clash Of Clans Hack Tool]

Recoil - Revision history

98.230.20.101: rm vandalism

en>Monkbot: /* Misconceptions about recoil */Fix CS1 deprecated date parameter errors

en>ZéroBot: r2.7.1) (Robot: Adding ca:Retrocés