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
thumb|An object is injective if, given a monomorphism : → , any : → can be extended to .
An object <math>Q</math> in a category <math>\mathbf{C}</math> is said to be injective if for every monomorphism <math>f: X \to Y</math> and every morphism <math>g: X \to Q</math> there exists a morphism <math>h: Y \to Q</math> extending <math>g</math> to <math>Y</math>, i.e. such that <math> h \circ f = g</math>.
That is, every morphism <math>X \to Q</math> factors through every monomorphism <math>X \hookrightarrow Y</math>.
The morphism <math>h</math> in the above definition is not required to be uniquely determined by <math>f</math> and <math>g</math>.
In a locally small category, it is equivalent to require that the hom functor <math>\operatorname{Hom}_{\mathbf{C(-,Q)</math> carries monomorphisms in <math>\mathbf{C}</math> 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 <math>\mathbf{C}</math> is an abelian category, an object Q of <math>\mathbf{C}</math> is injective if and only if its hom functor Hom<sub>C</sub>(–,Q) is exact.
If <math>0 \to Q \to U \to V \to 0</math> is an exact sequence in <math>\mathbf{C}</math> such that Q is injective, then the sequence splits.
Enough injectives and injective hulls
The category <math>\mathbf{C}</math> is said to have enough injectives if for every object X of <math>\mathbf{C}</math>, there exists a monomorphism from X to an injective object.
A monomorphism g in <math>\mathbf{C}</math> 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.