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.

is said to be injective if for every monomorphism

factors through every monomorphism

in the above definition is not required to be uniquely determined by

In a locally small category, it is equivalent to require that the hom functor

carries monomorphisms in

to surjective set maps.

The notion of injectivity was first formulated for abelian categories, and this is still one of its primary areas of application.

is an abelian category, an object Q of

is injective if and only if its hom functor HomC(–,Q) is exact.

is an exact sequence in

such that Q is injective, then the sequence splits.

is said to have enough injectives if for every object X of

, there exists a monomorphism from X to an injective object.

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.

[1] 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.

be a class of morphisms of

there exists a morphism

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

-essential if for any morphism f, the composite fg is in

-essential morphism with domain X and an

-injective codomain G, then G is called an