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.

General Definition

Let ${\mathfrak {C}}$ be a category and let ${\mathcal {H}}$ be a class of morphisms of ${\mathfrak {C}}$.

An object $Q$ of ${\mathfrak {C}}$ is said to be ${\mathcal {H}}$-injective if for every morphism $f:A\to Q$ and every morphism $h:A\to B$ in ${\mathcal {H}}$ there exists a morphism $g:B\to Q$ extending (the domain of) $f$, i.e. $gh=f$. In other words, $Q$ is injective iff any ${\mathcal {H}}$-morphism into $Q$ extends (via composition on the left) to a morphism into $Q$.

The morphism $g$ in the above definition is not required to be uniquely determined by $h$ and $f$.

In a locally small category, it is equivalent to require that the hom functor $Hom_{\mathfrak {C}}(-,Q)$ carries ${\mathcal {H}}$-morphisms to epimorphisms (surjections).

The classical choice for ${\mathcal {H}}$ is the class of monomorphisms, in this case, the expression injective object is used.

Additive category

Let

$0\to A\to B\to C\to 0$

be an exact sequence in an additive category such that A is injective. Then the sequence splits and B is injective if and only if C is injective.

Abelian case

If ${\mathfrak {C}}$ is an abelian category, an object A of ${\mathfrak {C}}$ is injective iff its hom functor Hom_C(–,A) is exact.

The abelian case was the original framework for the notion of injectivity.

Enough injectives

Let ${\mathfrak {C}}$ be a category, H a class of morphisms of ${\mathfrak {C}}$ ; the category ${\mathfrak {C}}$ is said to have enough H-injectives if for every object X of ${\mathfrak {C}}$, there exist a H-morphism from X to an H-injective object.

Injective hull

A H-morphism g in ${\mathfrak {C}}$ is called H-essential if for any morphism f, the composite fg is in H only if f is in H.

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).

References

• J. Rosicky, Injectivity and accessible categories
• F. Cagliari and S. Montovani, T₀-reflection and injective hulls of fibre spaces