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 $mathfrak{C}$ 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 monomorphisms, 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 its hom 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 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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