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.

General Definition

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

An object of is said to be -injective if for every morphism and every morphism in there exists a morphism extending (the domain of) , i.e. . In other words, is injective iff any -morphism into extends (via composition on the left) to a morphism into .

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

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.

Abelian case

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

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

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.

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 noncanonical isomorphism.



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

de:Injektiver Modul#Injektive Moduln