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