# Injective object

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{{Short description|Mathematical object in category theory}}
{{No footnotes|date=October 2021}}
In [mathematics](/source/mathematics), especially in the field of [category theory](/source/category_theory), the concept of '''injective object'''<ref>{{Cite web |title=Injective object - Encyclopedia of Mathematics |url=https://encyclopediaofmath.org/wiki/Injective_object |access-date=2025-07-28 |website=encyclopediaofmath.org}}</ref> is a generalization of the concept of [injective module](/source/injective_module). This concept is important in [cohomology](/source/cohomology), in [homotopy theory](/source/homotopy_theory) and in the theory of [model categories](/source/model_category). The dual notion is that of a [projective object](/source/projective_object).

==Definition==
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](/source/Object_(category_theory)) <math>Q</math> in a [category](/source/Category_(mathematics)) <math>\mathbf{C}</math> is said to be '''injective''' if for every [monomorphism](/source/monomorphism) <math>f: X \to Y</math> and every [morphism](/source/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 [locally small](/source/Category_(mathematics)) category, it is equivalent to require that the [hom functor](/source/hom_functor) <math>\operatorname{Hom}_{\mathbf{C}}(-,Q)</math> carries monomorphisms in <math>\mathbf{C}</math> to [surjective](/source/surjective) set maps.

==In Abelian categories==
The notion of injectivity was first formulated for [abelian categories](/source/Abelian_category), 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](/source/if_and_only_if) its [hom functor](/source/hom_functor) Hom<sub>'''C'''</sub>(&ndash;,''Q'') is [exact](/source/exact_functor).

If <math>0 \to Q \to U \to V \to 0</math> is an [exact sequence](/source/exact_sequence) in <math>\mathbf{C}</math> such that ''Q'' is injective, then the [sequence splits](/source/splitting_lemma).

==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](/source/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](/source/up_to) a non-canonical isomorphism.<ref name=":0" />

==Examples==
*In the category of [abelian group](/source/abelian_group)s and [group homomorphism](/source/group_homomorphism)s, '''Ab''', an injective object is necessarily a [divisible group](/source/divisible_group). Assuming the axiom of choice, the notions are equivalent.
*In the category of (left) [modules](/source/Module_(mathematics)) and [module homomorphism](/source/module_homomorphism)s, ''R''-'''Mod''', an injective object is an [injective module](/source/injective_module). ''R''-'''Mod''' has [injective hull](/source/injective_hull)s (as a consequence, ''R''-'''Mod''' has enough injectives).
*In the [category of metric spaces](/source/category_of_metric_spaces), '''Met''', an injective object is an [injective metric space](/source/injective_metric_space), and the injective hull of a metric space is its [tight span](/source/tight_span).
*In the category of [T<sub>0</sub> space](/source/T0_space)s and [continuous mapping](/source/continuous_mapping)s, an injective object is always a [Scott topology](/source/Scott_topology) on a [continuous lattice](/source/continuous_lattice), and therefore it is always [sober](/source/Sober_space) and [locally compact](/source/locally_compact).

==Uses==
If an abelian category has enough injectives, we can form [injective resolutions](/source/Injective_resolution), 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](/source/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](/source/Ext_functor), and [Tor](/source/Tor_functor) functors and also the various [cohomology](/source/cohomology) theories in [group theory](/source/group_theory), [algebraic topology](/source/algebraic_topology) and [algebraic geometry](/source/algebraic_geometry). The categories being used are typically [functor categories](/source/functor_category) or categories of [sheaves of ''O''<sub>''X''</sub> modules](/source/sheaf_of_modules) over some [ringed space](/source/ringed_space) (''X'', ''O''<sub>''X''</sub>) or, more generally, any [Grothendieck category](/source/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}} [factors through](/source/List_of_mathematical_jargon) {{var|h}}.]]

Let <math>\mathbf{C}</math> be a category and let <math>\mathcal{H}</math> be a [class](/source/Class_(set_theory)) 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](/source/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](/source/simplicial_set)s, the injective objects with respect to the class ''<math>\mathcal{H}</math>'' of anodyne extensions are [Kan complex](/source/Kan_complex)es.
*In the category of [partially ordered set](/source/partially_ordered_set)s and [monotone map](/source/monotone_map)s, the [complete lattice](/source/complete_lattice)s form the injective objects for the class ''<math>\mathcal{H}</math>'' of [order-embedding](/source/order-embedding)s, and the [Dedekind–MacNeille completion](/source/Dedekind%E2%80%93MacNeille_completion) of a partially ordered set is its ''<math>\mathcal{H}</math>''-injective hull.

==See also==
*[Projective object](/source/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](/source/de%3AInjektiver_Modul)

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Adapted from the Wikipedia article [Injective object](https://en.wikipedia.org/wiki/Injective_object) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Injective_object?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.