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 thereof.

General definition

Let $C$ be a category and $J\subset \mathrm{Mor}(C)$ a class of morphisms in $C$.

Example

Frequently $J$ is the class of all monomorphisms or a related class.

This is notably the case for $C$ is a category of chain complexes equipped with the injective model structure on chain complexes and $J$ is its class of cofibrations.

Definition

An object $I$ in $C$ is $J$-injective if all diagrams of the form

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

admit an extension

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

If $J$ is the class of all monomorphisms, we speak merely of an injective object.

Definition

Ones says that a category $C$ has enough injectives if every object admits a monomorphism into an injective object.

The dual notion is a projective object.

Assuming the axiom of choice, we have the following easy result.

Proposition

An arbitrary small product of injective objects is injective.

Remark

If $C$ has a terminal object $*$ then these extensions are equivalently lifts

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

and hence the $J$-injective objects are precisely those that have the right lifting property against the class $J$.

Remark

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.

In abelian categories

The term injective object is used most frequently in the context that $C$ is an abelian category.

Observation

For $C$ an abelian category the class $J$ of monomorphisms is the same as the class of morphisms $f:A\to B$ such that $0\to A\stackrel{f}{\to }B$ is exact.

Proof

By definition of abelian category every monomorphism $A\hookrightarrow B$ is a kernel, hence a pullback of the form

$$\begin{array}{ccc}A& \to & 0\\ \downarrow & & \downarrow \\ B& \to & C\end{array}$$\array{ A &\to& 0 \\ \downarrow && \downarrow \\ B &\to& C }

for $0$ the (algebraic) zero object. By the pasting law for pullbacks we find that also the left square in

$$\begin{array}{ccccc}0& \to & A& \to & 0\\ \downarrow & & \downarrow & & \downarrow \\ 0& \to & B& \to & C\end{array}$$\array{ 0 &\to& A &\to& 0 \\ \downarrow && \downarrow && \downarrow \\ 0 &\to& B &\to& C }

is a pullback, hence $0\to A\to B$ is exact.

Corollary

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^{\mathrlap{k}} & \swarrow_{\mathrlap{\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

Injective modules

Let $R$ be a commutative ring and $C=R\mathrm{Mod}$ the category of $R$-modules. We discuss injective modules over $R$ (see there for more).

The following criterion says that for identifying injective modules it is sufficient to test the right lifting property which characterizes injective objects by def. 1, only on those monomorphisms which include an ideal into the base ring $R$.

Proposition

(Baer's criterion)

If the axiom of choice holds, then a module $Q\in R\mathrm{Mod}$ is an injective module precisely if for $I$ any left $R$-ideal regarded as an $R$-module, any homomorphism $g:I\to Q$ in $C$ can be extended to all of $R$ along the inclusion $I\hookrightarrow R$.

This is due to (Baer).

Sketch of proof

Let $i:M\hookrightarrow N$ be a monomorphism in $R\mathrm{Mod}$, and let $f:M\to Q$ be a map. We must extend $f$ to a map $h:N\to Q$. Consider the poset whose elements are pairs $(M\prime ,f\prime )$ where $M\prime$ is an intermediate submodule between $M$ and $N$ and $f\prime :M\prime \to Q$ is an extension of $f$, ordered by $(M\prime ,f\prime )\le (M″,f″)$ if $M″$ contains $M\prime$ and $f″$ extends $f\prime$. By an application of Zorn's lemma, this poset has a maximal element, say $(M\prime ,f\prime )$. Suppose $M\prime$ is not all of $N$, and let $x\in N$ be an element not in $M\prime$; we show that $f\prime$ extends to a map $M″=⟨x⟩+M\prime \to Q$, a contradiction.

The set $\{r\in R:rx\in M\prime \}$ is an ideal $I$ of $R$, and we have a module homomorphism $g:I\to Q$ defined by $g(r)=f\prime (rx)$. By hypothesis, we may extend $g$ to a module map $k:R\to Q$. Writing a general element of $M″$ as $rx+y$ where $y\in M\prime$, it may be shown that

$$f″(rx+y)=k(r)+g(y)$$f''(r x + y) = k(r) + g(y)

is well-defined and extends $f\prime$, as desired.

Corollary

(Assume that the axiom of choice holds.) Let $R$ be a Noetherian ring, and let $\{Q_j{\}}_{j\in J}$ be a collection of injective modules over $R$. Then the direct sum $Q={⨁}_{j\in J}{Q}_j$ is also injective.

Proof

By Baer’s criterion, theorem 1, it suffices to show that for any ideal $I$ of $R$, a module homomorphism $f:I\to Q$ extends to a map $R\to Q$. Since $R$ is Noetherian, $I$ is finitely generated as an $R$-module, say by elements $x_1,\dots ,x_n$. Let $p_j:Q\to Q_j$ be the projection, and put $f_j=p_j\circ f$. Then for each $x_i$, $f_j(x_i)$ is nonzero for only finitely many summands. Taking all of these summands together over all $i$, we see that $f$ factors through

$$\prod _{j\in J\prime }{Q}_j=\underset{j\in J\prime }{⨁}{Q}_j\hookrightarrow Q$$\prod_{j \in J'} Q_j = \bigoplus_{j \in J'} Q_j \hookrightarrow Q

for some finite $J\prime \subset J$. But a product of injectives is injective, hence $f$ extends to a map $R\to {\prod }_{j\in J\prime }{Q}_j$, which completes the proof.

Proposition

Conversely, $R$ is a Noetherian ring if direct sums of injective $R$-modules are injective.

This is due to Bass and Papp. See (Lam, Theorem 3.46).

Injective abelian groups

Let $C=\mathbb{Z}\mathrm{Mod}\simeq$ Ab be the abelian category of abelian groups.

Proposition

If the axiom of choice holds, then an abelian group $A$ is an injective object in Ab precisely if it is a divisible group, in that for all integers $n\in \mathbb{N}$ we have $nG=G$.

This follows for instance using Baer's criterion, prop. 1.

An explit proof is spelled out at Planet math – abelian group is divisible if and only if it is an injective object

Example

By prop. 3 the following abelian groups are injective in Ab.

The group of rational numbers $\mathbb{Q}$ is injective in Ab, as is the additive group of real numbers $ℝ$ and generally that underlying any field. The additive group underlying any vector space is injective. The quotient of any injective group by any other group is injective.

Example

Not injective in Ab is for instance the cyclic group $\mathbb{Z}/n\mathbb{Z}$ for $n>1$.

Properties

Existence of enough injectives

We discuss a list of classes of categories that have enough injective according to def. 2.

Proposition

Every topos has enough injectives.

Proof

Every power object can be shown to be injective, and every object embeds into its power object by the “singletons” map.

Proposition

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.

Proposition

As soon as the category Ab of abelian groups has enough injectives, so does the abelian category $R$Mod of modules over some ring $R$.

Proof

Observe that the forgetful functor $U:R\mathrm{Mod}\to \mathrm{AbGp}$ has both a left adjoint $R_{!}$ (extension of scalars from $\mathbb{Z}$ to $ℝ$) and a right adjoint $R_{*}$ (coextension of scalars). Since it has a left adjoint, it is exact, and so its right adjoint $R_{*}$ preserves injective objects. Thus given any $R$-module $M$, we can embed $U(M)$ in an injective abelian group $I$, and then $M$ embeds in $R_{*}(I)$.

Proposition

For $R=k$ a field, hence $R$Mod = $k$Vect, ever object is both injective as well as projective.

Proposition

The category of abelian sheaves $\mathrm{Ab}(\mathrm{Sh}(C))$ on any small site $C$, hence the category of abelian groups in the sheaf topos over $C$, has enough injectives.

A proof of can be found in Peter Johnstone’s book Topos Theory, p261.

Remark

This is in stark contrast to the situation for projective objects, which generally do not exist in categories of sheaves.

Corollary

The category of sheaves of modules over any sheaf of rings? on any small site also enough injectives.

Proof

Combining prop. 8 with prop. 6 (which relativizes to any topos).

This slick proof of this important fact was pointed out by Colin McLarty in an email to the categories list dated 10 Oct 2010.

Injective resolutions

Proposition

Let $𝒜$ be an abelian category. Then for every object $X\in 𝒜$ there is an injective resolution, hence a chain complex

$$J^{•}=[J^0\to \cdots \to J^n\to \cdots ]\in {\mathrm{Ch}}_{(}𝒜)$$J^\bullet = [J^0 \to \cdots \to J^n \to \cdots] \in Ch_(\mathcal{A})

equipped with a a quasi-isomorphism of cochain complexes $X\stackrel{\sim }{\to }{J}^{•}$

$$\begin{array}{ccccccccc}X& \to & 0& \to & \cdots & \to & 0& \to & \cdots \\ \downarrow & & \downarrow & & & & \downarrow \\ J^0& \to & J^1& \to & codt& \to & J^n& \to & \cdots \end{array}.$$\array{ X &\to& 0 &\to& \cdots &\to& 0 &\to& \cdots \\ \downarrow && \downarrow && && \downarrow \\ J^0 &\to& J^1 &\to& \codt &\to& J^n &\to& \cdots } \,.

Related concepts

• projective object, projective presentation, projective cover, projective resolution

  • projective module

• injective object, injective presentation, injective envelope, injective resolution

  • injective module

• free object, free resolution

  • free module

• flat object, flat resolution

  • flat module

References

The notion of injective modules was introduced in

• R. Baer (1940)

(The dual notion of projective modules was considered explicitly only much later.)

A general discussion can be found in

• Masaki Kashiwara, Pierre Schapira, Categories and Sheaves

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

Using tools from the theory of accessible categories, injective objects are discussed in

• Jiri Rosicky, Injectivity and accessible categories (ps)

Baer’s criterion is discussed in many texts, for example

• N. Jacobsen, Basic Algebra II, W.H. Freeman and Company, 1980.

See also

• T.-Y. Lam, Lectures on modules and rings, Graduate Texts in Mathematics 189, Springer Verlag (1999).