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| 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]].
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| ==Formal definition==
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| === Algebraic objects ===
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| 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.).
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| 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:
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| # <math>f_{ii}\,</math> is the identity of <math>A_i\,</math>, and
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| # <math>f_{ik}= f_{jk}\circ f_{ij}</math> for all <math>i\le j\le k</math>.
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| Then the pair <math>\langle A_i,f_{ij}\rangle</math> is called a direct system over <math>I\,</math>.
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| 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>
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| 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>.
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| 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>.
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| 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.
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| An important property is that taking direct limits in the category of [[module (mathematics)|modules]] is an [[exact functor]].
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| ===Direct limit over a direct system in a category===
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| 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
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| <div style="text-align: center;">[[Image:DirectLimit-01.png]]</div>
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| [[commutative diagram|commute]] for all ''i'', ''j''. The direct limit is often denoted
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| :<math>X = \varinjlim X_i</math>
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| with the direct system <math>\langle X_i, f_{ij}\rangle</math> being understood.
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| 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.
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| 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]].
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| ==Examples==
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| *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>.
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| *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.
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| *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>).
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| *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''.
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| *Direct limits in the [[category of topological spaces]] are given by placing the [[final topology]] on the underlying set-theoretic direct limit.
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| *Direct limits are linked to [[inverse limit]]s via
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| :<math>\mathrm{Hom} (\varinjlim X_i, Y) = \varprojlim \mathrm{Hom} (X_i, Y).</math>
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| *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.
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| ==Related constructions and generalizations==
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| 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.
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| ==See also==
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| * [[Inverse limit|Inverse, or projective limit]]
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| ==References==
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| * {{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}}.
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| * {{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}}.
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| {{DEFAULTSORT:Direct Limit}}
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| [[Category:Limits (category theory)]]
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| [[Category:Abstract algebra]]
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