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 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. 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.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 is exact . If is an exact sequence in such that Q is injective, then the sequence splits .Enough injectives and injective hulls 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.Examples 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 modules and module homomorphisms, R -Mod , an injective object is an injective module. R -Mod has injective hulls. In the category of metric spaces , 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 .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 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 or, more generally, any Grothendieck category .Generalization 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 {{math|}}-injective objects
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