nLab
injective object

Definition

There is a very general notion of injective objects in a category $C$, and a sequence of refinements as $C$ is equipped with more structure and property, in particular for $C$ an abelian category or a relative.

General definition

Let $C$ be a category and $J$ a collection of morphisms in $C$. Frequently $J$ is the class of all monomorphisms or a related class. An object $I$ in $C$ is $J$-injective if all diagrams

$$\begin{array}{ccc}X& \to & I\\ {\downarrow }^{j\in J}\\ Z\end{array}$$\array{ X &\to& I \\ \downarrow^{j \in J} \\ Z }

admit an extension

$$\begin{array}{ccc}X& \to & I\\ {\downarrow }^{j\in J}& {\nearrow }_{\exists }\\ Z\end{array}.$$\array{ X &\to& I \\ \downarrow^{j \in J} & \nearrow_{\exists} \\ Z } \,.

If $C$ has a terminal object $*$ this can be thought of as a lift

$$\begin{array}{ccc}X& \to & I\\ {\downarrow }^{j\in J}& {\nearrow }_{\exists }& \downarrow \\ Z& \to & *\end{array}$$\array{ X &\to& I \\ \downarrow^{j \in J} & \nearrow_{\exists} & \downarrow \\ Z &\to& * }

as for factorization systems.

If $C$ is a locally small category then $I$ is $J$-injective precisely if the hom-functor

$${\mathrm{Hom}}_C(-,I):C^{\mathrm{op}}\to \mathrm{Set}$$Hom_C(-,I) : C^{op} \to Set

takes morphisms in $J$ to epimorphisms in Set.

If $J$ is the class of all monomorphisms, we speak merely of an injective object. We say that a category $C$ has enough injectives if every object admits a monomorphism into an injective object.

The dual notion is a projective object.

In abelian categories

The term injective object is used most frequently in the context that $C$ is an abelian category. In this case the class $J$ of monomorphisms is the same as the class of morphisms $f:X\to Y$ such that $0\to X\stackrel{f}{\to }Y$ is exact. An object $I$ of an abelian category $C$ is then injective if it satisfies the following equivalent conditions:

• the hom-functor ${\mathrm{Hom}}_C(-,I):C^{\mathrm{op}}\to \mathrm{Set}$ is exact;

• for all morphisms $f:X\to Y$ such that $0\to X\to Y$ is exact and for all $k:X\to I$, there exists $h:Y\to I$ such that $h\circ f=k$.

$$\begin{array}{ccccc}0& \to & X& \stackrel{f}{\to }& Y\\ & & {\downarrow }^k& {\swarrow }_{\exists h}\\ & & I\end{array}.$$\array{ 0 &\to& X &\stackrel{f}{\to}& Y \\ && \downarrow^k & \swarrow_{\exists h} \\ && I } \,.

In chain complexes

See homotopically injective object for a relevant generalization to categories of chain complexes, and its relationship to ordinary injectivity.

Examples

• Every topos has enough injectives. In fact, every power object can be shown to be injective, and every object embeds into its power object by the “singletons” map.

• At least assuming some form of the axiom of choice, the category of abelian groups has enough injectives. Full AC is much more than required, however; small violations of choice suffices. The abelian category of modules over some ring is similar.

• The category of abelian sheaves on any small site also has enough injectives. This is in stark contrast to the situation for projectives, which generally do not exist in categories of sheaves.

References

Much of this discussion can be found in

• Kashiwara–Schapira, Categories and Sheaves

The general notion of injective objects is in section 9.5, the case of injective complexes in section 14.1.