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 homotopy theory and in theory of model categories. The dual notion is that of a projective object.
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 extending (the domain of) , i.e. .
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 -morphisms to epimorphisms (surjections).
The classical choice for is the class of monomorphisms, in this case, the expression injective object is used.
The abelian case was the original framework for the notion of injectivity (and still the most important one). If is an abelian category, an object A of is injective iff its hom functor HomC(–,A) is exact.
be an exact sequence in such that A is injective. Then the sequence splits and B is injective if and only if C is injective.
Let be a category, H a class of morphisms of ; the category is said to have enough H-injectives if for every object X of , there exist a H-morphism from X to an H-injective object. Again, H is often the class of monomorphisms, and the classical definition of having enough injectives is that for every object X of , there exist a monomorphism from X to an injective object.
A H-morphism g in is called H-essential if for any morphism f, the composite fg is in H only if f is in H. If H is the class of monomorphisms, g is called an essential monomorphism.
If f is a H-essential H-morphism with a domain X and an H-injective codomain G, G is called an H-injective hull of X. This H-injective hull is then unique up to a noncanonical isomorphism.
- In the category of Abelian groups and group homomorphisms, an injective object is a divisible group.
- In the category of 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 category of metric spaces and nonexpansive mappings, 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 therefore it is always sober and locally compact.
- In the category of simplicial sets, the injective objects with respect to the class of anodyne extensions are Kan complexes.
- In the category of partially ordered sets and monotonic functions between posets, the complete lattices form the injective objects for order-embeddings, and the Dedekind–MacNeille completion of a partially ordered set is its injective hull.
- One also talks about injective objects in more general categories, for instance in functor categories or in categories of sheaves of OX modules over some ringed space (X,OX).
- Proof: Since the sequence splits, B is a direct sum of A and C.
- J. Rosicky, Injectivity and accessible categories
- F. Cagliari and S. Montovani, T0-reflection and injective hulls of fibre spaces