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 cohomology, in homotopy theory and in the theory of model categories. The dual notion is that of a projective object.


An object Q is injective if, given a monomorphism f : XY, any g : XQ can be extended to Y.

An object   in a category   is said to be injective if for every monomorphism   and every morphism   there exists a morphism   extending   to  , i.e. such that  .

That is, every morphism   factors through every monomorphism  .

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 monomorphisms in   to surjective set maps.

In Abelian categoriesEdit

The notion of injectivity was first formulated for abelian categories, and this is still one of its primary areas of application. When   is an abelian category, an object Q of   is injective if and only if its hom functor HomC(–,Q) is exact.

If   is an exact sequence in   such that Q is injective, then the sequence splits.

Enough injectives and injective hullsEdit

The category   is said to have enough injectives if for every object X of  , there exists a monomorphism from X to an injective object.

A monomorphism g in   is called an essential monomorphism if for any morphism f, the composite fg is a monomorphism only if f is a monomorphism.

If g is an essential monomorphism with domain X and an injective codomain G, then G is called an injective hull of X. The injective hull is then uniquely determined by X up to a non-canonical isomorphism.



If an abelian category has enough injectives, we can form injective resolutions, i.e. for a given object X we can form a long exact sequence


and one can then define the derived functors of a given functor F by applying F to this sequence and computing the homology of the resulting (not necessarily exact) sequence. This approach is used to define Ext, and Tor functors and also the various cohomology theories in group theory, algebraic topology and algebraic geometry. The categories being used are typically functor categories or categories of sheaves of OX modules over some ringed space (X, OX) or, more generally, any Grothendieck category.


An object Q is H-injective if, given h : AB in H, any f : AQ factors through h.

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   with  .

If   is the class of monomorphisms, we are back to the injective objects that were treated above.

The category   is said to have enough  -injectives if for every object X of  , there exists an  -morphism from X to an  -injective object.

A  -morphism g in   is called  -essential if for any morphism f, the composite fg is in   only if f is in  .

If g is a  -essential morphism with domain X and an  -injective codomain G, then G is called an  -injective hull of X.

Examples of H-injective objectsEdit

See alsoEdit



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