Injective object

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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.

Contents 1 General Definition 2 Abelian case 3 Enough injectives 4 Injective hull 5 Examples 6 References

[edit] General Definition

Let be a category and let be a class of morphisms of .

An object Q of is said to be -injective if every arrow and every morphisms in there exists a morphism extending f, i.e gh = f. In other words, Q is injective iff any -morphism extends to any morphism into Q.

The morphism g in the above definition is not required to be uniquely determined by h and f.

In a locally small category, it is equivalent to require that the hom functor carries -morphisms to epimorphisms (surjections).

The classical choice for is the class of monomorphisms, in this case, the expression injective object is used.

[edit] Abelian case

If is an abelian category, an object A of is injective iff its hom functor Hom_C(–,A) is exact.

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

[edit] Enough injectives

Let be a category, H a class of morphisms of  ; the category is said to have enough H-injectives if for every object X of , there exist a H-morphism from X to an H-injective object.

[edit] Injective hull

A H-morphism g in 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.

[edit] 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).

[edit] References

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