
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.
Definition.
An object formula_1 in a category formula_2 is said to be injective if for every monomorphism formula_3 and every morphism formula_4 there exists a morphism formula_5 extending formula_6 to formula_7, i.e. such that formula_8.
That is, every morphism formula_9 factors through every monomorphism formula_10.
The morphism formula_11 in the above definition is not required to be uniquely determined by formula_12 and formula_6.
In a locally small category, it is equivalent to require that the hom functor formula_14 carries monomorphisms in formula_2 to surjective set maps.
In Abelian categories.
The notion of injectivity was first formulated for abelian categories, and this is still one of its primary areas of application. When formula_2 is an abelian category, an object "Q" of formula_2 is injective if and only if its hom functor HomC(–,"Q") is exact.
If formula_18 is an exact sequence in formula_2 such that "Q" is injective, then the sequence splits.
Enough injectives and injective hulls.
The category formula_2 is said to "have enough injectives" if for every object "X" of formula_2, there exists a monomorphism from "X" to an injective object. 
A monomorphism "g" in formula_2 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.
Uses.
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 "O""X" modules over some ringed space ("X", "O""X") or, more generally, any Grothendieck category.
Generalization.
Let formula_2 be a category and let formula_25 be a class of morphisms of formula_2.
An object formula_1 of formula_2 is said to be "formula_25"-injective if for every morphism formula_30 and every morphism formula_31 in formula_25 there exists a morphism formula_33 with formula_34.
If formula_25 is the class of monomorphisms, we are back to the injective objects that were treated above.
The category formula_2 is said to "have enough formula_25-injectives" if for every object "X" of formula_2, there exists an "formula_25"-morphism from "X" to an "formula_25"-injective object.
A "formula_25"-morphism "g" in formula_2 is called "formula_25"-essential if for any morphism "f", the composite "fg" is in "formula_25" only if "f" is in "formula_25".
If "g" is a "formula_25"-essential morphism with domain "X" and an "formula_25"-injective codomain "G", then "G" is called an formula_25-injective hull of "X". 