# 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 homotopy theory and in theory of model categories. The dual notion is that of a projective object.

## Definition

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.

## Abelian case

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 Hom_{C}(–,*A*) is exact.

Let

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.^{[1]}

## Enough injectives

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 every object *X* of , there exist a monomorphism from *X* to an injective object.

## Injective hull

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.

## Examples

- 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 O
_{X}modules over some ringed space (*X*,O_{X}).

## See also

## Notes

- ↑ Proof: Since the sequence splits,
*B*is a direct sum of*A*and*C*.

## References

- J. Rosicky, Injectivity and accessible categories
- F. Cagliari and S. Montovani, T
_{0}-reflection and injective hulls of fibre spaces