Injective object

In mathematics, especially in the field of , 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 . The dual notion is that of a projective object. An in a 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 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 , 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. 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. In the category of abelian groups and group homomorphisms, Ab, an injective object is necessarily a divisible group. Assuming the axiom of choice, the notions are equivalent. In the category of (left) modules and module homomorphisms, R-Mod, an injective object is an injective module. R-Mod has injective hulls (as a consequence, R-Mod has enough injectives). In the , Met, an injective object is an injective metric space, and the injective hull of a metric space is its tight span. In the category of T0 spaces and continuous mappings, an injective object is always a Scott topology on a continuous lattice, and therefore it is always sober and locally compact.