# Injective object

Injective object

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 categories. The dual notion is that of a projective object.

General Definition

Let $mathfrak\left\{C\right\}$ be a category and let $mathcal\left\{H\right\}$ be a class of morphisms of $mathfrak\left\{C\right\}$.

An object $Q$ of $mathfrak\left\{C\right\}$ is said to be "$mathcal\left\{H\right\}$"-injective if every arrow $f: A o Q$ and every morphisms $h: A o B$ in $mathcal\left\{H\right\}$ there exists a morphism $g: B o Q$ extending $f$, i.e $gh = f$. In other words, $Q$ is injective iff any $mathcal\left\{H\right\}$-morphism extends to any morphism into $Q$.

The morphism $g$ in the above definition is not required to be uniquely determined by $h ext\left\{ and \right\} f$.

In a locally small category, it is equivalent to require that the hom functor $Hom_\left\{mathfrak\left\{C\left(-,Q\right)$ carries $mathcal\left\{H\right\}$-morphisms to epimorphisms (surjections).

The classical choice for $mathcal\left\{H\right\}$ is the class of monomorphisms, in this case, the expression injective object is used.

Abelian case

If $mathfrak\left\{C\right\}$ is an abelian category, an object "A" of $mathfrak\left\{C\right\}$ is injective iff its hom functor HomC(&ndash;,"A") is exact.

The abelian case was the original framework for the notion of injectivity.

Enough injectives

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

Injective hull

A "H"-morphism "g" in $mathfrak\left\{C\right\}$ 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 groups and group homomorphisms, an injective object is a divisible group.
*In the category of modules and module homomorphisms, "R"-Mod, an injective object is an injective module. "R"-Mod has injective hulls (as a consequence, R-Mod has enough injectives).
*In the category of metric spaces and nonexpansive mappings, an injective object is an injective metric space.
*In the category of T0 spaces and continuous mappings, an injective object is always a Scott topology on a continuous lattice therefore it is always sober and locally compact.
*In the category of simplicial sets, the injective objects with respect to the class of anodyne extensions are Kan complexes.
*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 some ringed space ("X",O"X").

References

*J. Rosicky, Injectivity and accessible categories
*F. Cagliari and S. Montovani, T0-reflection and injective hulls of fibre spaces

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