Editing Injective object

{{Short description|Mathematical object in category theory}}
{{No footnotes|date=October 2021}}
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 [[cohomology]], in [[homotopy theory]] and in the theory of [[model category|model categories]]. The dual notion is that of a [[projective object]].

==Definition==
[[File:Diagrammadefinizionemoduloiniettivo.png|thumb|An object {{var|Q}} is injective if, given a monomorphism {{var|f}} : {{var|X}} → {{var|Y}}, any {{var|g}} : {{var|X}} → {{var|Q}} can be extended to {{var|Y}}.]]

An [[Object (category theory)|object]] <math>Q</math> in a [[Category (mathematics)|category]] <math>\mathbf{C}</math> is said to be '''injective''' if for every [[monomorphism]] <math>f: X \to Y</math> and every [[morphism]] <math>g: X \to Q</math> there exists a morphism <math>h: Y \to Q</math> extending <math>g</math> to <math>Y</math>, i.e. such that <math> h \circ f = g</math>.<ref name=":0">{{Cite book |last=Adamek |first=Jiri |url=http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf |title=Abstract and Concrete Categories: The Joy of Cats |last2=Herrlich |first2=Horst |last3=Strecker |first3=George |publisher=Reprints in Theory and Applications of Categories, No. 17 (2006) pp. 1-507. orig. John Wiley |year=1990 |pages=147-155 |chapter=Sec. 9. Injective objects and essential embeddings}}</ref>

That is, every morphism <math>X \to Q</math> factors through every monomorphism <math>X \hookrightarrow Y</math>.

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

In a [[Category_(mathematics)#Small_and_large_categories|locally small]] category, it is equivalent to require that the [[hom functor]] <math>\operatorname{Hom}_{\mathbf{C}}(-,Q)</math> carries monomorphisms in <math>\mathbf{C}</math> to [[surjective]] set maps.

==In Abelian categories==
The notion of injectivity was first formulated for [[Abelian category|abelian categories]], and this is still one of its primary areas of application. When <math>\mathbf{C}</math> is an abelian category, an object ''Q'' of <math>\mathbf{C}</math> is injective [[if and only if]] its [[hom functor]] Hom<sub>'''C'''</sub>(&ndash;,''Q'') is [[exact functor|exact]].

If <math>0 \to Q \to U \to V \to 0</math> is an [[exact sequence]] in <math>\mathbf{C}</math> such that ''Q'' is injective, then the [[splitting lemma|sequence splits]].

==Enough injectives and injective hulls==
The category <math>\mathbf{C}</math> is said to ''have enough injectives'' if for every object ''X'' of <math>\mathbf{C}</math>, there exists a monomorphism from ''X'' to an injective object. 

A monomorphism ''g'' in <math>\mathbf{C}</math> is called an [[essential monomorphism]] if for any morphism ''f'', the composite ''fg'' is a monomorphism only if ''f'' is a monomorphism.

If ''g'' is an essential monomorphism with domain ''X'' and an injective codomain ''G'', then ''G'' is called an '''injective hull''' of ''X''.  The injective hull is then uniquely determined by ''X'' [[up to]] a non-canonical isomorphism.<ref name=":0" />

==Examples==
*In the category of [[abelian group]]s and [[group homomorphism]]s, '''Ab''', an injective object is necessarily a [[divisible group]]. Assuming the axiom of choice, the notions are equivalent.
*In the category of (left) [[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 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|T<sub>0</sub> space]]s and [[continuous mapping]]s, an injective object is always a [[Scott topology]] on a [[continuous lattice]], and therefore it is always [[Sober space|sober]] and [[locally compact]].

==Uses==
If an abelian category has enough injectives, we can form [[Injective resolution|injective resolutions]], i.e. for a given object ''X'' we can form a long exact sequence
:<math>0\to X \to Q^0 \to Q^1 \to Q^2 \to \cdots</math>
and one can then define the [[derived functor]]s of a given functor ''F'' by applying ''F'' to this sequence and computing the homology of the resulting (not necessarily exact) sequence. This approach is used to define [[Ext functor|Ext]], and [[Tor functor|Tor]] functors and also the various [[cohomology]] theories in [[group theory]], [[algebraic topology]] and [[algebraic geometry]]. The categories being used are typically [[functor category|functor categories]] or categories of [[sheaf of modules|sheaves of ''O''<sub>''X''</sub> modules]] over some [[ringed space]] (''X'', ''O''<sub>''X''</sub>) or, more generally, any [[Grothendieck category]].

== Generalization ==
[[File:Injective object.svg|thumb|An object {{var|Q}} is {{var|H}}-injective if, given {{var|h}} : {{var|A}} → {{var|B}} in {{var|H}}, any {{var|f}} : {{var|A}} → {{var|Q}} [[List of mathematical jargon#factor through|factors through]] {{var|h}}.]]

Let <math>\mathbf{C}</math> be a category and let <math>\mathcal{H}</math> be a [[Class (set theory)|class]] of morphisms of <math>\mathbf{C}</math>.

An object <math>Q</math> of <math>\mathbf{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> with <math> g \circ h = f</math>.

If <math>\mathcal{H}</math> is the class of [[monomorphism]]s, we are back to the injective objects that were treated above.

The category <math>\mathbf{C}</math> is said to ''have enough <math>\mathcal{H}</math>-injectives'' if for every object ''X'' of <math>\mathbf{C}</math>, there exists an ''<math>\mathcal{H}</math>''-morphism from ''X'' to an ''<math>\mathcal{H}</math>''-injective object.

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

If ''g'' is a ''<math>\mathcal{H}</math>''-essential morphism with domain ''X'' and an ''<math>\mathcal{H}</math>''-injective codomain ''G'', then ''G'' is called an '''<math>\mathcal{H}</math>-injective hull''' of ''X''.<ref name=":0" /> 

=== Examples of {{math|{{mathcal|H}}}}-injective objects===

*In the category of [[simplicial set]]s, the injective objects with respect to the class ''<math>\mathcal{H}</math>'' of anodyne extensions are [[Kan complex]]es.
*In the category of [[partially ordered set]]s and [[monotone map]]s, the [[complete lattice]]s form the injective objects for the class ''<math>\mathcal{H}</math>'' of [[order-embedding]]s, and the [[Dedekind–MacNeille completion]] of a partially ordered set is its ''<math>\mathcal{H}</math>''-injective hull.

==See also==
*[[Projective object]]

==Notes==
{{reflist}}

==References==
*Jiri Adamek, Horst Herrlich, George Strecker. Abstract and concrete categories: The joy of cats, Chapter 9, Injective Objects and Essential Embeddings, [http://www.tac.mta.ca/tac/reprints/articles/17/tr17.pdf Republished in Reprints and Applications of Categories, No. 17 (2006) pp. 1-507], Wiley (1990).
*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]]