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.


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

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

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 every object X of , there exist a monomorphism from X to an 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 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


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


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