Injective object

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 ${\displaystyle \mathbf{C}}$ is an abelian category, an object Q of ${\displaystyle \mathbf{C}}$ is injective if and only if its hom functor Hom_C(–,Q) is exact.

If ${\displaystyle 0\to Q\to U\to V\to 0}$ is an exact sequence in ${\displaystyle \mathbf{C}}$ such that Q is injective, then the sequence splits.

Enough injectives and injective hulls

The category ${\displaystyle \mathbf{C}}$ is said to have enough injectives if for every object X of ${\displaystyle \mathbf{C}}$, there exists a monomorphism from X to an injective object.

A monomorphism g in ${\displaystyle \mathbf{C}}$ 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 (left) 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, 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

${\displaystyle 0\to X\to Q^0\to Q^1\to Q^2\to \cdots }$

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 O_X modules over some ringed space (X, O_X) or, more generally, any Grothendieck category.

Generalization

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

An object ${\displaystyle Q}$ of ${\displaystyle \mathbf{C}}$ is said to be ${\displaystyle \mathcal{H}}$-injective if for every morphism ${\displaystyle f:A\to Q}$ and every morphism ${\displaystyle h:A\to B}$ in ${\displaystyle \mathcal{H}}$ there exists a morphism ${\displaystyle g:B\to Q}$ with ${\displaystyle g\circ h=f}$.

If ${\displaystyle \mathcal{H}}$ is the class of monomorphisms, we are back to the injective objects that were treated above.

The category ${\displaystyle \mathbf{C}}$ is said to have enough ${\displaystyle \mathcal{H}}$-injectives if for every object X of ${\displaystyle \mathbf{C}}$, there exists an ${\displaystyle \mathcal{H}}$-morphism from X to an ${\displaystyle \mathcal{H}}$-injective object.

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

If g is a ${\displaystyle \mathcal{H}}$-essential morphism with domain X and an ${\displaystyle \mathcal{H}}$-injective codomain G, then G is called an ${\displaystyle \mathcal{H}}$-injective hull of X.

Examples of H-injective objects

• In the category of simplicial sets, the injective objects with respect to the class ${\displaystyle \mathcal{H}}$ of anodyne extensions are Kan complexes.
• In the category of partially ordered sets and monotone maps, the complete lattices form the injective objects for the class ${\displaystyle \mathcal{H}}$ of order-embeddings, and the Dedekind–MacNeille completion of a partially ordered set is its ${\displaystyle \mathcal{H}}$-injective hull.