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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{injective object} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{category_theory}{}\paragraph*{{Category theory}}\label{category_theory} [[!include category theory - contents]] \hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra} [[!include homological algebra - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{general_definition}{General definition}\dotfill \pageref*{general_definition} \linebreak \noindent\hyperlink{in_abelian_categories}{In abelian categories}\dotfill \pageref*{in_abelian_categories} \linebreak \noindent\hyperlink{in_chain_complexes}{In chain complexes}\dotfill \pageref*{in_chain_complexes} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{InjectiveModules}{Injective modules}\dotfill \pageref*{InjectiveModules} \linebreak \noindent\hyperlink{injective_abelian_groups}{Injective abelian groups}\dotfill \pageref*{injective_abelian_groups} \linebreak \noindent\hyperlink{in_toposes}{In toposes}\dotfill \pageref*{in_toposes} \linebreak \noindent\hyperlink{in_topological_spaces}{In topological spaces}\dotfill \pageref*{in_topological_spaces} \linebreak \noindent\hyperlink{in_boolean_algebras}{In Boolean algebras}\dotfill \pageref*{in_boolean_algebras} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{preservation_of_injective_objects}{Preservation of injective objects}\dotfill \pageref*{preservation_of_injective_objects} \linebreak \noindent\hyperlink{ExistenceOfEnoughInjectives}{Existence of enough injectives}\dotfill \pageref*{ExistenceOfEnoughInjectives} \linebreak \noindent\hyperlink{InjectiveResolutions}{Injective resolutions}\dotfill \pageref*{InjectiveResolutions} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} There is a very general notion of \emph{injective objects} in a [[category]] $C$, and then there is a sequence of more concrete notions as $C$ is equipped with more [[stuff, structure, property|structure and property]], in particular for $C$ an [[abelian category]] [[additive and abelian categories|or a relative]] thereof. The concept of [[resolutions]] by injective objects -- [[injective resolutions]] -- is crucial notably in the discussion of [[derived functors]] (in the context of [[abelian categories]]: [[derived functors in homological algebra]]). Being injective is a \emph{property} of an object; the corresponding \emph{structure} is called an [[algebraic injective]]. \hypertarget{general_definition}{}\subsubsection*{{General definition}}\label{general_definition} Let $C$ be a [[category]] and $J \subset Mor(C)$ a [[class]] of [[morphisms]] in $C$. \begin{example} \label{}\hypertarget{}{} 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]]. \end{example} \begin{defn} \label{InjectiveObjects}\hypertarget{InjectiveObjects}{} An object $I$ in $C$ is \textbf{$J$-injective} if all diagrams of the form \begin{displaymath} \itexarray{ X &\to& I \\ {}^{\mathllap{j \in J}}\downarrow \\ Z } \end{displaymath} admit an [[extension]] \begin{displaymath} \itexarray{ X &\to& I \\ {}^{\mathllap{j \in J}}\downarrow & \nearrow_{\mathrlap{\exists}} \\ Z } \,. \end{displaymath} If $J$ is the class of all [[monomorphisms]], we speak merely of an \textbf{injective object}. \end{defn} \begin{remark} \label{}\hypertarget{}{} If $C$ has a [[terminal object]] $\ast$, then the $J$-injective objects $I$ according to def. \ref{InjectiveObjects} are those for which $I \stackrel{\exists!}{\to} \ast$ is called a $J$-[[injective morphisms]]. \end{remark} \begin{defn} \label{EnoughInjectives}\hypertarget{EnoughInjectives}{} Ones says that a category $C$ has \textbf{enough injectives} if every object admits a [[monomorphism]] into an injective object. \end{defn} The dual notion is a [[projective object]]. Assuming the axiom of choice, we have the following easy result. \begin{prop} \label{}\hypertarget{}{} An arbitrary small product of injective objects is injective. \end{prop} \begin{remark} \label{}\hypertarget{}{} If $C$ has a [[terminal object]] $*$ then these extensions are equivalently lifts \begin{displaymath} \itexarray{ X &\to& I \\ {}^{\mathllap{j \in J}}\downarrow & \nearrow_{\mathrlap{\exists}} & \downarrow \\ Z &\to& * } \end{displaymath} and hence the $J$-injective objects are precisely those that have the [[right lifting property]] against the class $J$. \end{remark} \begin{remark} \label{}\hypertarget{}{} If $C$ is a [[locally small category]] then $I$ is $J$-injective precisely if the [[hom-functor]] \begin{displaymath} Hom_C(-,I) : C^{op} \to Set \end{displaymath} takes morphisms in $J$ to [[epimorphism]]s in [[Set]]. \end{remark} \hypertarget{in_abelian_categories}{}\subsubsection*{{In abelian categories}}\label{in_abelian_categories} The term \emph{injective object} is used most frequently in the context that $C$ is an [[abelian category]]. \begin{prop} \label{}\hypertarget{}{} 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 sequence|exact]]. \end{prop} \begin{proof} By definition of [[abelian category]] every monomorphism $A \hookrightarrow B$ is a [[kernel]], hence a [[pullback]] of the form \begin{displaymath} \itexarray{ A &\to& 0 \\ \downarrow && \downarrow \\ B &\to& C } \end{displaymath} for $0$ the (algebraic) [[zero object]]. By the for pullbacks we find that also the left square in \begin{displaymath} \itexarray{ 0 &\to& A &\to& 0 \\ \downarrow && \downarrow && \downarrow \\ 0 &\to& B &\to& C } \end{displaymath} is a pullback, hence $0 \to A \to B$ is exact. \end{proof} \begin{cor} \label{EquivalentCharacterizationOfInjectivesInAbelianCategories}\hypertarget{EquivalentCharacterizationOfInjectivesInAbelianCategories}{} An [[object]] $I$ of an abelian category $C$ is then \textbf{injective} if it satisfies the following equivalent conditions: \begin{itemize}% \item the [[hom-functor]] $Hom_C(-, I) : C^{op} \to Set$ is [[exact functor|exact]]; \item for all [[morphism]]s $f : X \to Y$ such that $0 \to X \to Y$ is [[exact sequence|exact]] and for all $k : X \to I$, there exists $h : Y \to I$ such that $h\circ f = k$. \end{itemize} \begin{displaymath} \itexarray{ 0 &\to& X &\stackrel{f}{\to}& Y \\ && \downarrow^{\mathrlap{k}} & \swarrow_{\mathrlap{\exists h}} \\ && I } \,. \end{displaymath} \end{cor} By the [[formal dual]] of \href{projective+module#NProjectiveIFFHomNExact}{this prop.}. \hypertarget{in_chain_complexes}{}\subsubsection*{{In chain complexes}}\label{in_chain_complexes} See [[homotopically injective object]] for a relevant generalization to categories of chain complexes, and its relationship to ordinary injectivity. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \hypertarget{InjectiveModules}{}\subsubsection*{{Injective modules}}\label{InjectiveModules} Let $R$ be a [[commutative ring]] and $C = R Mod$ 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. \ref{InjectiveObjects}, only on those [[monomorphisms]] which include an [[ideal]] into the base ring $R$. \begin{theorem} \label{BaerTheorem}\hypertarget{BaerTheorem}{} \textbf{([[Baer's criterion]])} If the [[axiom of choice]] holds, then a [[module]] $Q \in R Mod$ 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 $I \hookrightarrow R$. \end{theorem} This is due to (\hyperlink{Baer}{Baer}). \begin{proof} Let $i \colon M \hookrightarrow N$ be a [[monomorphism]] in $R Mod$, and let $f \colon M \to Q$ be a map. We must extend $f$ to a map $h \colon N \to Q$. Consider the [[poset]] whose elements are pairs $(M', f')$ where $M'$ is an intermediate [[submodule]] between $M$ and $N$ and $f' \colon M' \to Q$ is an extension of $f$, ordered by $(M', f') \leq (M'', f'')$ if $M''$ contains $M'$ and $f''$ extends $f'$. By an application of [[Zorn's lemma]], this poset has a [[maximal element]], say $(M', f')$. Suppose $M'$ is not all of $N$, and let $x \in N$ be an element not in $M'$; we show that $f'$ extends to a map $M'' = \langle x \rangle + M' \to Q$, a [[contradiction]]. The set $\{r \in R: r x \in M'\}$ is an ideal $I$ of $R$, and we have a module [[homomorphism]] $g \colon I \to Q$ defined by $g(r) = f'(r x)$. By [[hypothesis]], we may extend $g$ to a module map $k \colon R \to Q$. Writing a general element of $M''$ as $r x + y$ where $y \in M'$, it may be shown that \begin{displaymath} f''(r x + y) = k(r) + f'(y) \end{displaymath} is well-defined and extends $f'$, as desired. \end{proof} \begin{corollary} \label{DirectSumInjectives}\hypertarget{DirectSumInjectives}{} (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]] $Q = \bigoplus_{j \in J} Q_j$ is also injective. \end{corollary} \begin{proof} By Baer's criterion, theorem \ref{BaerTheorem}, it suffices to show that for any [[ideal]] $I$ of $R$, a module [[homomorphism]] $f \colon I \to Q$ extends to a map $R \to Q$. Since $R$ is Noetherian, $I$ is [[finitely generated module|finitely generated]] as an $R$-module, say by elements $x_1, \ldots, x_n$. Let $p_j \colon Q \to Q_j$ 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 \begin{displaymath} \prod_{j \in J'} Q_j = \bigoplus_{j \in J'} Q_j \hookrightarrow Q \end{displaymath} for some finite $J' \subset J$. But a [[product]] of injectives is injective, hence $f$ extends to a map $R \to \prod_{j \in J'} Q_j$, which completes the proof. \end{proof} \begin{prop} \label{}\hypertarget{}{} Conversely, $R$ is a [[Noetherian ring]] if [[direct sums]] of injective $R$-[[modules]] are injective. \end{prop} This is due to Bass and Papp. See (\hyperlink{Lam}{Lam, Theorem 3.46}). \hypertarget{injective_abelian_groups}{}\subsubsection*{{Injective abelian groups}}\label{injective_abelian_groups} Let $C = \mathbb{Z} Mod \simeq$ [[Ab]] be the [[abelian category]] of [[abelian groups]]. \begin{prop} \label{InjectiveAbelianGroupIsDivisibleGroup}\hypertarget{InjectiveAbelianGroupIsDivisibleGroup}{} 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 $n G = G$. \end{prop} This follows for instance using [[Baer's criterion]], prop. \ref{BaerTheorem}. An explicit proof is spelled out at \href{http://planetmath.org/AbelianGroupIsDivisibleIfAndOnlyIfItIsAnInjectiveObject.html}{Planet math -- abelian group is divisible if and only if it is an injective object} \begin{example} \label{}\hypertarget{}{} By prop. \ref{InjectiveAbelianGroupIsDivisibleGroup} 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]] $\mathbb{R}$ and generally that underlying any [[field]] of characteristic zero. The additive group underlying any [[vector space]] is injective. The [[quotient]] of any injective group by any other group is injective. \end{example} \begin{example} \label{}\hypertarget{}{} \textbf{Not} injective in [[Ab]] is for instance the [[cyclic group]] $\mathbb{Z}/n\mathbb{Z}$ for $n \gt 1$. \end{example} \hypertarget{in_toposes}{}\subsubsection*{{In toposes}}\label{in_toposes} In any [[topos]], the [[subobject classifier]] $\Omega$ is an injective object, as is any [[powering|power]] of $\Omega$ ([[Mac Lane-Moerdijk]], IV.10). Also one can define various notions of \emph{internally} injective objects. These turn out to be equivalent: \begin{prop} \label{EquivalenceOfInternalNotionsOfInjectivity}\hypertarget{EquivalenceOfInternalNotionsOfInjectivity}{} In any [[elementary topos]] $\mathcal{E}$ with a [[natural numbers object]], the following statements about an object $I \in \mathcal{E}$ are equivalent. \begin{enumerate}% \item The functor $[-, I] : \mathcal{E}^op \to \mathcal{E}$ maps monomorphisms in $\mathcal{E}$ to epimorphisms. \item The functor $[-, I] : \mathcal{E}^op \to \mathcal{E}$ maps monomorphisms in $\mathcal{E}$ to morphisms for which any global element of the target locally (after [[change of base]] along an epimorphism) possesses a preimage. \item For any morphism $p : A \to 1$ in $\mathcal{E}$, the object $p^*I$ has property 1. as an object of $\mathcal{E}/A$. \item For any morphism $p : A \to 1$ in $\mathcal{E}$, the object $p^*I$ has property 2. as an object of $\mathcal{E}/A$. \item The interpretation of the statement ``$I$ is an injective object'' using the [[stack semantics]] of $\mathcal{E}$ holds. \end{enumerate} \end{prop} \begin{proof} The implications ``1. $\Rightarrow$ 2.'', ``3 $\Rightarrow$ 4.'', ``3. $\Rightarrow$ 1.'', and ``4. $\Rightarrow$ 2.'' are trivial. The equivalence ``3. $\Leftrightarrow$ 5.'' follows directly from the interpretation rules of the stack semantics. The implication ``2. $\Rightarrow$ 4.'' employs the [[base change\#GeometricMorphism|extra left-adjoint]] $p_! : \mathcal{E}/A \to \mathcal{E}$ to $p^* : \mathcal{E} \to \mathcal{E}/A$, as in the usual proof that injective sheaves remain injective when restricted to smaller open subsets: We have that $p_* \circ [-, p^*I]_{\mathcal{E}/A} \cong [-, I]_{\mathcal{E}} \circ p_!$, the functor $p_!$ preserves monomorphisms, and one can check that $p_*$ reflects the property that global elements locally possess preimages. Details are in (\hyperlink{Harting}{Harting, Theorem 1.1}). The implication ``4. $\Rightarrow$ 3.'' follows by performing an extra change of base, since any non-global element becomes a global element after a suitable change of base. \end{proof} Somewhat surprisingly, and in stark contrast with the situation for [[internally projective objects]], internal injectivity coincides with external injecticity. \begin{prop} \label{EquivalenceOfInternalAndExternalInjectivity}\hypertarget{EquivalenceOfInternalAndExternalInjectivity}{} Let $\mathcal{E}$ be the topos of [[sheaves]] over a [[locale]]. Then an object $I \in \mathcal{E}$ is internally injective (in any of the senses given by Proposition \ref{EquivalenceOfInternalNotionsOfInjectivity}) if and only if $I$ is injective as in Definition \ref{InjectiveObjects}. \end{prop} \begin{proof} Let $I$ be an externally injective object. Then $I$ satisfies condition 2. of Proposition \ref{EquivalenceOfInternalNotionsOfInjectivity}, even without having to pass to a cover. Conversely, let $I$ be an internally injective object. Let $m : X \to Y$ be a monomorphism and let $k : X \to I$ be an arbitrary morphism. We want to show that there exists an extension $Y \to I$ of $k$ along $m$. To this end, consider the sheaf \begin{displaymath} F \coloneqq \{ k' : \mathcal{H}om(Y,I) | k' \circ m = k \}. \end{displaymath} One can check that $F$ is [[flabby sheaf|flabby]] (this is particularly easy using the [[internal language]], details will be added later) and therefore has a global section. \end{proof} \begin{remark} \label{}\hypertarget{}{} The analogs of Proposition \ref{EquivalenceOfInternalNotionsOfInjectivity} and Proposition \ref{EquivalenceOfInternalAndExternalInjectivity} for abelian group objects instead of unstructured objects hold as well, with mostly the same proofs. Condition 1. then refers to the functor $[-, X] : Ab(\mathcal{E})^op \to Ab(\mathcal{E})$. \end{remark} \hypertarget{in_topological_spaces}{}\subsubsection*{{In topological spaces}}\label{in_topological_spaces} In the category [[Top]] of all topological spaces, the injective objects are precisely the [[inhabited set|inhabited]] [[indiscrete spaces]]. In the category of $T_0$ spaces (see [[separation axiom]]), the injective objects are the terminal spaces. In the above two cases, this refers to injectivity with respect to monomorphisms. In the category of $T_0$ spaces, the injective objects with respect to homeomorphic embeddings are precisely those given by [[Scott topologies]] on [[continuous lattices]]; as [[locales]] these are [[local compactum|locally compact]] and [[spatial locale|spatial]]. (Such spaces are usually called, perhaps confusingly, \emph{injective spaces}.) In the category of all spaces, the injectives with respect to homeomorphic embeddings (i.e., [[regular monomorphisms]]) are the spaces whose $T_0$ reflections are continuous lattices under the Scott topology. \hypertarget{in_boolean_algebras}{}\subsubsection*{{In Boolean algebras}}\label{in_boolean_algebras} Injective objects in the category of [[Boolean algebras]] are precisely [[complete Boolean algebras]]. This is the dual form of a theorem of Gleason, saying that the [[projective objects]] in the category of [[Stone spaces]] are the [[extremally disconnected topological space|extremally disconnected]] ones (the closure of every open set is again open). \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{preservation_of_injective_objects}{}\subsubsection*{{Preservation of injective objects}}\label{preservation_of_injective_objects} \begin{lemma} \label{RightAdjointsOfExactFunctorsPreserveInjectives}\hypertarget{RightAdjointsOfExactFunctorsPreserveInjectives}{} Given a pair of [[additive functor|additive]] [[adjoint functors]] \begin{displaymath} (L \dashv R) \;\colon\; \mathcal{B} \stackrel{\overset{L}{\longleftarrow}}{\underset{R}{\longrightarrow}} \mathcal{A} \end{displaymath} between [[abelian categories]] such that the [[left adjoint]] $L$ is a [[left exact functor]] (thus automatically exact), then the [[right adjoint]] preserves injective objects. \end{lemma} \begin{proof} Observe that an object is injective precisely if the [[hom-functor]] into it sends [[monomorphisms]] to [[epimorphisms]], and that $L$ preserves monomorphisms by assumption of (left-)exactness. With this the statement follows via adjunction isomorphism \begin{displaymath} Hom_{\mathcal{A}}(-,R(I))\simeq Hom_{\mathcal{B}}(L(-),I) \,. \end{displaymath} \end{proof} \begin{remark} \label{}\hypertarget{}{} Additivity of the left adjoint follows from the remaining assumptions, since [[additive functor\#SufficientConditions|exact functors preserve biproducts]]. \end{remark} The preceding lemma has the following variant: \begin{prop} \label{Adjuncts_Injectives}\hypertarget{Adjuncts_Injectives}{} Let $\mathcal{C}$, $\mathcal{D}$ be categories and $L\dashv R:\mathcal{D}\to\mathcal{C}$ be an adjunction. If $L$ maps monos to monos, then $R$ maps injectives to injectives. \end{prop} \begin{proof} Let $I\in\mathcal{D}$ be injective. Consider the following diagram in $\mathcal{C}$: \begin{displaymath} \itexarray{ A & \overset{m}{\rightarrowtail} & B \\ f\downarrow & & \\ R(I) & & } \end{displaymath} Let $\theta: Hom_\mathcal{C}(X,R(Y))\overset{\simeq}{\rightarrow}Hom_\mathcal{D}(L(X),Y)$ the natural bijection given by the adjunction. Consider now the following diagram in $\mathcal{D}$ where the assumptions ensure that $L(m)$ is mono: \begin{displaymath} \itexarray{ L(A) & \overset{L(m)}{\rightarrowtail} & L(B) \\ \theta(f)\downarrow & & \\ I & & } \end{displaymath} Since $I$ is injective, there there exists a filler $\theta(g):L(B)\to I$ which by the adjunction must come from a uniquely determined $g:B\to R(I)$. But the naturality of the bijection with respect to composition says that \begin{displaymath} \frac{L(A)\overset{L(m)}{\rightarrowtail} L(B)\overset{\theta(g)}{\rightarrow}I}{A\overset{m}{\rightarrowtail} B\overset{g}{\rightarrow}R(I)} \end{displaymath} correspond to each other under the bijection whence $\theta(g\circ m)=\theta(g)\circ L(m)$ but from the commutativity of the second diagram we have $\theta(g)\circ L(m)=\theta(f)=\theta(g\circ m)$. Since $\theta$ is a bijection it follows that $f=g\circ m$ which proves that $R(I)$ is injective. \end{proof} \begin{remark} \label{}\hypertarget{}{} The proof transposes the proof of the dual statement 10.2. in Hilton-Stammbach (\hyperlink{HS71}{1971}, p.82): In situation $L\dashv R$, if $R$ maps epis to epis then $L$ maps projectives to projectives. \end{remark} \begin{remark} \label{Exponential_injectives}\hypertarget{Exponential_injectives}{} Let $X\in\mathcal{C}$ such that ${}_-\times:\mathcal{C}\to\mathcal{C}$ exists and has a right adjoint ${}_-\times X\dashv (_-)^X$. Since it is easy to check that ${}_-\times X$ preserves monos it follows that $(_-)^X$ preserves injectives. In particular, for toposes this implies that all power objects $\Omega^X$ are injective since the injectivity of $\Omega$ follows more or less straightforwardly from its classifying properties. \end{remark} \hypertarget{ExistenceOfEnoughInjectives}{}\subsubsection*{{Existence of enough injectives}}\label{ExistenceOfEnoughInjectives} We discuss a list of classes of categories that have \emph{enough injectives} according to def. \ref{EnoughInjectives}. \begin{prop} \label{}\hypertarget{}{} Every [[topos]] has enough injectives. \end{prop} \begin{proof} Every [[power object]] can be shown to be injective (cf. the above \hyperlink{Exponentials_injectives}{remark}), and every object embeds into its power object by the ``singletons'' map. \end{proof} \begin{prop} \label{AbHasEnoughInjectives}\hypertarget{AbHasEnoughInjectives}{} Assuming some form of the [[axiom of choice]], the category of [[abelian groups]] has enough injectives. \end{prop} Full AC is much more than required, however; [[small violations of choice]] suffices. \begin{proof} By prop. \ref{InjectiveAbelianGroupIsDivisibleGroup} an [[abelian group]] is an injective $\mathbb{Z}$-module precisely if it is a [[divisible group]]. So we need to show that every [[abelian group]] is a [[subgroup]] of a [[divisible group]]. To start with, notice that the group $\mathbb{Q}$ of [[rational numbers]] is divisible and hence the canonical embedding $\mathbb{Z} \hookrightarrow \mathbb{Q}$ shows that the additive group of [[integers]] embeds into an injective $\mathbb{Z}$-module. Now by the discussion at \emph{[[projective module]]} every [[abelian group]] $A$ receives an [[epimorphism]] $(\oplus_{s \in S} \mathbb{Z}) \to A$ from a [[free group|free]] abelian group, hence is the [[quotient group]] of a [[direct sum]] of copies of $\mathbb{Z}$. Accordingly it embeds into a quotient $\tilde A$ of a direct sum of copies of $\mathbb{Q}$. \begin{displaymath} \itexarray{ ker &\stackrel{=}{\to}& ker \\ \downarrow && \downarrow \\ (\oplus_{s \in S} \mathbb{Z}) &\hookrightarrow& (\oplus_{s \in S} \mathbb{Q}) \\ \downarrow && \downarrow \\ A &\hookrightarrow& \tilde A } \end{displaymath} Here $\tilde A$ is divisible because the [[direct sum]] of divisible groups is again divisible, and also the [[quotient group]] of a divisible groups is again divisble. So this exhibits an embedding of any $A$ into a divisible abelian group, hence into an injective $\mathbb{Z}$-module. \end{proof} \begin{lemma} \label{TransferOfEnoughInjectivesAlongAdjunctions}\hypertarget{TransferOfEnoughInjectivesAlongAdjunctions}{} Given a pair of [[additive functor|additive]] [[adjoint functors]] \begin{displaymath} (L \dashv R) \;\colon\; \mathcal{B} \stackrel{\overset{L}{\longleftarrow}}{\underset{R}{\longrightarrow}} \mathcal{A} \end{displaymath} between [[abelian categories]] such that the [[left adjoint]] $L$ is \begin{enumerate}% \item an [[exact functor]], \item a [[faithful functor]]. \end{enumerate} Then if $\mathcal{B}$ has enough injectives, also $\mathcal{A}$ has enough injectives. \end{lemma} \begin{proof} Consider $A \in \mathcal{A}$. By the assumption that $\mathcal{B}$ has enough injectives, there is an injective object $I \in \mathcal{B}$ and a monomorphism $i \colon L(A) \hookrightarrow I$. The [[adjunct]] of this is a morphism \begin{displaymath} \tilde i \colon A \longrightarrow R(I) \end{displaymath} and so it is sufficient to show that \begin{enumerate}% \item $R(I)$ is injective in $\mathcal{A}$; \item $\tilde i$ is a monomorphism. \end{enumerate} The first point is the statement of lemma \ref{RightAdjointsOfExactFunctorsPreserveInjectives}. For the second point, consider the [[kernel]] of $\tilde i$ as part of the [[exact sequence]] \begin{displaymath} ker(\tilde i)\longrightarrow A \stackrel{\tilde i}{\longrightarrow} R(I) \,. \end{displaymath} By the assumption that $L$ is an [[exact functor]], the image of this sequence under $L$ is still exact \begin{displaymath} L(ker(\tilde i)) \longrightarrow L(A) \stackrel{L(\tilde i)}{\longrightarrow} L(R(I)) \,. \end{displaymath} Now observe that $L(\tilde i)$ is a monomorphism: this is because its composite $L(A) \stackrel{L(\tilde i)}{\longrightarrow} L(R(I)) \stackrel{\epsilon}{\longrightarrow} I$ with the [[adjunction unit]] is (by the formula for [[adjuncts]]) the original morphism $i$, which by construction is a monomorphism. Therefore the exactness of the above sequence means that $L(ker(\tilde i)) \to L(A)$ is the [[zero morphism]]; and by the assumption that $L$ is a [[faithful functor]] this means that already $ker(\tilde i) \to A$ is zero, hence that $ker(\tilde i) = 0$, hence that $\tilde i$ is a monomorphism. \end{proof} \begin{prop} \label{RModHasEnoughInjectives}\hypertarget{RModHasEnoughInjectives}{} As soon as the category [[Ab]] of [[abelian groups]] has enough injectives, so does the [[abelian category]] $R$[[Mod]] of [[modules]] over some [[ring]] $R$. In particular if the [[axiom of choice]] holds, then $R Mod$ has enough injectives. \end{prop} \begin{proof} Observe that the [[forgetful functor]] $U\colon R Mod \to AbGp$ 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 functor|exact]]. Thus the statement follows via lemma \ref{TransferOfEnoughInjectivesAlongAdjunctions} from prop. \ref{AbHasEnoughInjectives}. \end{proof} \begin{prop} \label{}\hypertarget{}{} For $R = k$ a [[field]], hence $R$[[Mod]] = $k$[[Vect]], every object is both injective as well as [[projective object|projective]]. \end{prop} \begin{prop} \label{AbelianSheavesHaveEnoughProjectives}\hypertarget{AbelianSheavesHaveEnoughProjectives}{} The category of [[abelian sheaves]] $Ab(Sh(C))$ on any [[small site]] $C$, hence the category of abelian groups in the [[sheaf topos]] over $C$, has enough injectives. \end{prop} A proof of can be found in [[Peter Johnstone]]`s book \emph{Topos Theory}, p261. \begin{remark} \label{}\hypertarget{}{} This is in stark contrast to the situation for [[projective objects]], which generally do not exist in categories of sheaves. \end{remark} \begin{cor} \label{}\hypertarget{}{} The category of sheaves of modules over any [[sheaf of rings]] on any [[small site]] also enough injectives. \end{cor} \begin{proof} Combining prop. \ref{AbelianSheavesHaveEnoughProjectives} with prop. \ref{AbHasEnoughInjectives} (which relativizes to any topos). \end{proof} This slick proof of this important fact was pointed out by [[Colin McLarty]] in an email to the categories list dated 10 Oct 2010. \hypertarget{InjectiveResolutions}{}\subsubsection*{{Injective resolutions}}\label{InjectiveResolutions} \begin{prop} \label{}\hypertarget{}{} Let $\mathcal{A}$ be an [[abelian category]]. Then for every object $X \in \mathcal{A}$ there is an [[injective resolution]], hence a [[chain complex]] \begin{displaymath} J^\bullet = [J^0 \to \cdots \to J^n \to \cdots] \in Ch_(\mathcal{A}) \end{displaymath} equipped with a a [[quasi-isomorphism]] of [[cochain complexes]] $X \stackrel{\sim}{\to} J^\bullet$ \begin{displaymath} \itexarray{ X &\to& 0 &\to& \cdots &\to& 0 &\to& \cdots \\ \downarrow && \downarrow && && \downarrow \\ J^0 &\to& J^1 &\to& \cdots &\to& J^n &\to& \cdots } \,. \end{displaymath} \end{prop} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[projective object]], [[projective presentation]], [[projective cover]], [[projective resolution]] \begin{itemize}% \item [[projective module]] \end{itemize} \item \textbf{injective object}, [[injective presentation]], [[injective envelope]], [[injective resolution]] \begin{itemize}% \item [[injective module]] \end{itemize} \item [[algebraically injective object]] \item [[free object]], [[free resolution]] \begin{itemize}% \item [[free module]] \end{itemize} \item flat object, [[flat resolution]] \begin{itemize}% \item [[flat module]] \end{itemize} \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} The notion of [[injective modules]] was introduced in \begin{itemize}% \item R. Baer, \emph{Abelian groups that are direct summands of every containing abelian group} , Bulletin AMS \textbf{46} no. 10 (1940) pp.800-806. (\href{http://projecteuclid.org/euclid.bams/1183503234}{projecteuclid}) \end{itemize} (The dual notion of [[projective modules]] was considered explicitly only much later.) A general discussion can be found in \begin{itemize}% \item [[Masaki Kashiwara]], [[Pierre Schapira]], \emph{[[Categories and Sheaves]]} \end{itemize} 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 \begin{itemize}% \item [[Jiri Rosicky]], \emph{Injectivity and accessible categories} (\href{http://www.math.muni.cz/~rosicky/papers/acc5.ps}{ps}) \end{itemize} \hyperlink{Baer}{Baer's criterion} is discussed in many texts, for example \begin{itemize}% \item N. Jacobsen, \emph{Basic Algebra II}, W.H. Freeman and Company, 1980. \end{itemize} See also \begin{itemize}% \item T.-Y. Lam, \emph{Lectures on modules and rings}, Graduate Texts in Mathematics 189, Springer Verlag (1999). \end{itemize} For injective objects in a [[topos]] see \begin{itemize}% \item [[Francis Borceux]], \emph{Handbook of Categorical Algebra vol. 3} , Cambridge UP 1994. (section 5.6, pp.314-315) \item [[Saunders Mac Lane]], [[Ieke Moerdijk]], \emph{Sheaves in geometry and Logic} , Springer Heidelberg 1994. (section IV.10, pp.210-213) \item D. Higgs, \emph{Injectivity in the topos of complete Heyting algebra valued sets} , Can. J. Math. \textbf{36} (1984) pp.550-568. (\href{http://cms.math.ca/openaccess/cjm/v36/cjm1984v36.0550-0568.pdf}{pdf}) \item [[Peter Johnstone]], [[Fred Linton]], [[Robert Paré]], \emph{Injective Objects in Topoi II: Connections with the axiom of choice} , pp.207-216 in LNM \textbf{719} Springer Heidelberg 1979. \item T. Kenney, \emph{Injective Power Objects and the Axiom of Choice} , JPAA \textbf{215} (2011) pp.131--144. \item [[Fred Linton]], \emph{Injective Objects in Topoi III: Stability under coproducts} , Bull. Acad. Polon. Sci. S\'e{}r. Sci. Math. Astronom. Phys. \textbf{29} (1981) pp.341-347. \item [[Fred Linton]], [[Robert Paré]], \emph{Injective Objects in Topoi I: Representing coalgebras as algebras} , pp.196-206 in LNM \textbf{719} Springer Heidelberg 1979. \end{itemize} Discussion of injective objects ([[types]]) in [[homotopy type theory]]: \begin{itemize}% \item [[Martín Escardó]], \emph{Injectives types in univalent mathematics} (\href{https://arxiv.org/abs/1903.01211}{arXiv:1903.01211}) \end{itemize} For a detailed discussion of internal notions of injectivity see \begin{itemize}% \item Roswitha Harting, \emph{Locally injective abelian groups in a topos}, Communications in Algebra 11 (4), 1983. \end{itemize} For injective toposes in the 2-category of [[bounded topos|bounded toposes]] see \begin{itemize}% \item [[Peter Johnstone]], \emph{Injective Toposes} , pp.284-297 in LNM \textbf{871} Springer Heidelberg 1981. \item [[Peter Johnstone]], \emph{Sketches of an Elephant vol. 2} , Cambridge UP 2002. (section C4.3, pp.738-745) \end{itemize} For the proof of \hyperlink{Adjuncts_Injectives}{Lemma} we consulted \begin{itemize}% \item [[Peter Hilton]], Urs Stammbach, \emph{A Course in Homological Algebra} , GTM 4 Springer Heidelberg 1971. \end{itemize} [[!redirects injective objects]] [[!redirects enough injectives]] [[!redirects injective type]] [[!redirects injective types]] \end{document}