- Definition
- In Abelian categories
- Enough injectives and injective hulls
- Examples
- Uses
- Generalization Examples
- See also
- Notes
- References
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 the 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.
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 
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.
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, 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 (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.
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 OX modules over some ringed space (X,OX) or in general 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 exist a -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 H-injective hull of X.
Examples
- 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, the complete lattices form the injective objects for the class of order-embeddings, and the Dedekind–MacNeille completion of a partially ordered set is its -injective hull.
See also
- Projective object
Notes
References
- J. Rosicky, Injectivity and accessible categories
- F. Cagliari and S. Montovani, T0-reflection and injective hulls of fibre spaces
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