- Injective object
In

mathematics , especially in the field ofcategory theory , the concept of**injective object**is a generalization of the concept ofinjective module . This concept is important inhomotopy theory and in theory of model categories. The dual notion is that of aprojective object .**General Definition**Let $mathfrak\{C\}$ be a category and let $mathcal\{H\}$ be a class of morphisms of $mathfrak\{C\}$.

An object $Q$ of $mathfrak\{C\}$ is said to be

**"$mathcal\{H\}$"-injective**if every arrow $f:\; A\; o\; Q$ and every morphisms $h:\; A\; o\; B$ in $mathcal\{H\}$ there exists a morphism $g:\; B\; o\; Q$ extending $f$, i.e $gh\; =\; f$. In other words, $Q$ is injective iff any $mathcal\{H\}$-morphism extends to any morphism into $Q$.The morphism $g$ in the above definition is not required to be uniquely determined by $h\; ext\{\; and\; \}\; f$.

In a locally small category, it is equivalent to require that the

hom functor $Hom\_\{mathfrak\{C(-,Q)$ carries $mathcal\{H\}$-morphisms to epimorphisms (surjections).The classical choice for $mathcal\{H\}$ is the class of

monomorphism s, in this case, the expression**injective object**is used.**Abelian case**If $mathfrak\{C\}$ is an

abelian category , an object "A" of $mathfrak\{C\}$ is injective iff itshom functor Hom_{C}(–,"A") is exact.The abelian case was the original framework for the notion of injectivity.

**Enough injectives**Let $mathfrak\{C\}$ be a category, "H" a class of morphisms of $mathfrak\{C\}$ ; the category $mathfrak\{C\}$ is said to "have enough H-injectives" if for every object "X" of $mathfrak\{C\}$, there exist a "H"-morphism from "X" to an "H"-injective object.

**Injective hull**A "H"-morphism "g" in $mathfrak\{C\}$ 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 andgroup homomorphism s, an injective object is adivisible group .

*In the category of modules andmodule homomorphism s, "R"-Mod, an injective object is aninjective module . "R"-Mod hasinjective hull s (as a consequence, R-Mod has enough injectives).

*In the category ofmetric space s andnonexpansive mapping s, an injective object is aninjective metric space .

*In the category ofT0 space s andcontinuous mapping s, an injective object is always aScott topology on acontinuous lattice therefore it is always sober andlocally compact .

*In the category ofsimplicial set s, the injective objects with respect to the class of anodyne extensions areKan complex es.

*One also talks about injective objects in more general categories, for instance in functor categories or in categories of sheaves of O_{"X"}modules over someringed space ("X",O_{"X"}).**References***J. Rosicky, Injectivity and accessible categories

*F. Cagliari and S. Montovani, T_{0}-reflection and injective hulls of fibre spaces

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