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.


Q is H-injective if, given AB in H, any AQ extends to BQ.

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. .

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[edit]

The abelian case was the original framework for the notion of injectivity (and still the most important one). If is an abelian category, an object A of is injective iff its hom functor HomC(–,A) is exact.


be an exact sequence in such that A is injective. Then the sequence splits and B is injective if and only if C is injective.[1]

Enough injectives[edit]

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. Again, H is often the class of monomorphisms, and the classical definition of having enough injectives is that for every object X of , there exist a monomorphism from X to an injective object.

Injective hull[edit]

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 H is the class of monomorphisms, g is called an essential monomorphism.

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.


See also[edit]


  1. ^ Proof: Since the sequence splits, B is a direct sum of A and C.


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