and
nonabelian homological algebra
There is a very general notion of injective objects in a category $C$, and then there is a sequence of more concrete notions as $C$ is equipped with more structure and property, in particular for $C$ an abelian category 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).
Let $C$ be a category and $J \subset Mor(C)$ a class of morphisms in $C$.
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.
An object $I$ in $C$ is $J$-injective if all diagrams of the form
admit an extension
If $J$ is the class of all monomorphisms, we speak merely of an injective object.
If $C$ has a terminal object $\ast$, then the $J$-injective objects $I$ according to def. 1 are those for which $I \stackrel{\exists!}{\to} \ast$ is called a $J$-injective morphisms.
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.
An arbitrary small product of injective objects is injective.
If $C$ has a terminal object $*$ then these extensions are equivalently lifts
and hence the $J$-injective objects are precisely those that have the right lifting property against the class $J$.
If $C$ is a locally small category then $I$ is $J$-injective precisely if the hom-functor
takes morphisms in $J$ to epimorphisms in Set.
The term injective object is used most frequently in the context that $C$ is an abelian category.
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.
By definition of abelian category every monomorphism $A \hookrightarrow B$ is a kernel, hence a pullback of the form
for $0$ the (algebraic) zero object. By the pasting law for pullbacks we find that also the left square in
is a pullback, hence $0 \to A \to B$ is exact.
An object $I$ of an abelian category $C$ is then injective if it satisfies the following equivalent conditions:
the hom-functor $Hom_C(-, I) : C^{op} \to Set$ 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 $h\circ f = k$.
See homotopically injective object for a relevant generalization to categories of chain complexes, and its relationship to ordinary injectivity.
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. 1, only on those monomorphisms which include an ideal into the base ring $R$.
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 $I \hookrightarrow R$.
This is due to (Baer).
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
is well-defined and extends $f'$, as desired.
(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.
By Baer’s criterion, theorem 1, 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 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
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.
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).
Let $C = \mathbb{Z} Mod \simeq$ Ab be the abelian category of abelian groups.
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 $n G = G$.
This follows for instance using Baer's criterion, prop. 1.
An explicit proof is spelled out at Planet math – abelian group is divisible if and only if it is an injective object
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 $\mathbb{R}$ 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.
Not injective in Ab is for instance the cyclic group $\mathbb{Z}/n\mathbb{Z}$ for $n \gt 1$.
In any topos, the subobject classifier $\Omega$ is an injective object, as is any power of $\Omega$ (Mac Lane-Moerdijk, IV.10).
Also one can define various notions of internally injective objects. These turn out to be equivalent:
In any elementary topos $\mathcal{E}$ with a natural numbers object, the following statements about an object $I \in \mathcal{E}$ are equivalent.
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 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 (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.
Somewhat surprisingly, and in stark contrast with the situation for internally projective objects, internal injectivity coincides with external injecticity.
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 4) if and only if $I$ is injective as in Definition 1.
Let $I$ be an externally injective object. Then $I$ satisfies condition 2. of Proposition 4, 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
One can check that $F$ is flabby (this is particularly easy using the internal language, details will be added later) and therefore has a global section.
The analogs of Proposition 4 and Proposition 5 for abelian group objects instead of unstructured objects hold as well, with mostly the same proofs. Condition 1. then refers to the functor $[-, X] : Ab(\mathcal{E})^op \to Ab(\mathcal{E})$.
In the category Top of all topological spaces, the injective objects are precisely the inhabited indiscrete spaces.
In the category of $T_0$ spaces (see separation axiom), the injective objects are precisely those given by Scott topologies on continuous lattices; as locales these are locally compact and spatial.
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 ones (the closure of every open set is again open).
Given a pair of additive adjoint functors
between abelian categories such that the left adjoint $L$ is a left exact functor (thus automatically exact), then the right adjoint preserves injective objects.
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
Additivity of the left adjoint follows from the remaining assumptions, since exact functors preserve biproducts.
We discuss a list of classes of categories that have enough injective according to def. 2.
Every topos has enough injectives.
Every power object can be shown to be injective, and every object embeds into its power object by the “singletons” map.
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.
By prop. 3 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 projective module every abelian group $A$ receives an epimorphism $(\oplus_{s \in S} \mathbb{Z}) \to A$ from a 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 $\mathbb{Q}$.
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.
Given a pair of additive adjoint functors
between abelian categories such that the left adjoint $L$ is
an exact functor,
Then if $\mathcal{B}$ has enough injectives, also $\mathcal{A}$ has enough injectives.
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
and so it is sufficient to show that
$R(I)$ is injective in $\mathcal{A}$;
$\tilde i$ is a monomorphism.
The first point is the statement of lemma 1.
For the second point, consider the kernel of $\tilde i$ as part of the exact sequence
By the assumption that $L$ is an exact functor, the image of this sequence under $L$ is still exact
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.
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.
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. Thus the statement follows via lemma 2 from prop. 7.
For $R = k$ a field, hence $R$Mod = $k$Vect, every object is both injective as well as projective.
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.
A proof of can be found in Peter Johnstone’s book Topos Theory, p261.
This is in stark contrast to the situation for projective objects, which generally do not exist in categories of sheaves.
The category of sheaves of modules over any sheaf of rings? on any small site also enough injectives.
This slick proof of this important fact was pointed out by Colin McLarty in an email to the categories list dated 10 Oct 2010.
Let $\mathcal{A}$ be an abelian category. Then for every object $X \in \mathcal{A}$ there is an injective resolution, hence a chain complex
equipped with a a quasi-isomorphism of cochain complexes $X \stackrel{\sim}{\to} J^\bullet$
projective object, projective presentation, projective cover, projective resolution
injective object, injective presentation, injective envelope, injective resolution
flat object, flat resolution
The notion of injective modules was introduced in
(The dual notion of projective modules was considered explicitly only much later.)
A general discussion can be found in
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
Baer’s criterion is discussed in many texts, for example
See also
See
for a detailed discussion of internal notions of injectivity.