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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]].
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==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>(&ndash;,''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]]

Latest revision as of 00:39, 7 December 2014

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