Previously we proved a theorem due to Gabriel characterizing categories of modules as cocomplete abelian categories with a compact projective generator, where “generator” meant “every object is a colimit of finite direct sums of copies of the object.”
But we also used “generator” to mean “every object is a colimit of copies of the object,” and noted that these conditions are not equivalent: as this MO question discusses, the abelian group satisfies the first condition but not the second. More generally, as Mike Shulman explains here, there are in fact many inequivalent definitions of “generator” in category theory.
The goal of this post is to sort through a few of these definitions, which turn out to be totally ordered in strength, and find additional hypotheses under which they agree. As an application we’ll restate Gabriel’s theorem using weaker definitions of “generator” and give a more explicit description of all of the rings Morita equivalent to a given ring.
All categories appearing in this post will again be either “ordinary” (-enriched) or “linear” (-enriched). will again denote either presheaves or depending on whether is ordinary or linear.
A family of objects in a category is an essentially small full subcategory . This is partly just a way of saying “set of objects” which is invariant under equivalence, although the morphisms in will be important too. At a few points in this post we will take coproducts involving every object in . If is essentially small but not small these coproducts should be understood as ranging over representatives of the isomorphism classes of .
The naming convention for generation conditions is that for some adjective X, a single object is an “X generator” and a family of objects is a “family of X generators.” In both cases this might be shortened to “is X.” We will not use the convention that a family of objects is also an “X generator” because we want to be able to summarize Gabriel’s theorem using the phrase “compact projective generator.”
Two classical definitions
In this section we won’t make any explicit use of linearity: everything applies to linear categories thought of as ordinary categories.
Definition: A family of objects is a family of generators or faithful generators (nonstandard) if the following equivalent conditions hold:
- The functors are jointly faithful.
- The restricted Yoneda embedding is faithful.
- (If has coproducts) For all , the natural map is epic.
- (If has coproducts) For all , some map is epic.
Explicitly, these conditions mean that if is a pair of morphisms such that for all , then .
If is linear, #2 does not depend on whether we take set-valued or abelian group-valued presheaves. #3 and #4 are presumably the motivation for the term “generator”: if, for example, is a group and is a subset of , then generates iff the induced map is surjective.
Definition: A functor is conservative if it reflects isomorphisms: if is an isomorphism, then is an isomorphism.
Definition: A family of objects is a family of strong generators if the following equivalent conditions hold:
- The functors are jointly faithful and conservative.
- The restricted Yoneda embedding is faithful and conservative.
Explicitly, these conditions mean first that the above condition holds and second that if is a morphism such that is an isomorphism for all , then is an isomorphism.
Example. Let be a familiar category of algebraic objects such as groups, rings, or modules over a ring. Then the free object on one generator is a strong generator: is the forgetful functor, which is faithful by definition, and a morphism of groups, rings, or modules over a ring which is a bijection is an isomorphism. This generalizes to the category of models of any Lawvere theory.
Example. is a generator but not a strong generator: is the forgetful functor, which is faithful by definition, but a continuous bijection between topological spaces need not be a homeomorphism.
Example. (compact Hausdorff spaces) is a strong generator: is the forgetful functor, which is faithful by definition, and a continuous bijection between compact Hausdorff spaces is a homeomorphism. This generalizes to any category which is monadic over .
Recall that a basic property of faithful functors is that they reflect monos and epis. The corresponding property of conservative functors is the following.
Proposition: Let be a conservative functor. Then reflects any (shapes of) limits or colimits that it preserves.
Proof. By duality it suffices to prove the statement for colimits. Suppose that is a shape of colimit which have and which preserves. To say that reflects colimits of shape is to say the following: suppose is a cocone over a diagram of shape , so that it is equipped with suitable maps . If the induced maps exhibit as the colimit , then the original maps exhibit as the colimit .
Equivalently, the maps and describe maps and , and the condition is that if the latter map is an isomorphism, then so is the former map. But by hypothesis, since preserves colimits of shape , the former map is also the induced map , and if this is an isomorphism then, since is conservative, so is the map .
Corollary: Let be a conservative functor. If has either equalizers or coequalizers and preserves them, then is faithful.
Proof. Let be a pair of parallel arrows. Then iff is the coequalizer of iff is the equalizer of . If is conservative and preserves either of these, then it reflects them as well.
Corollary: If a category has equalizers, then we can drop “jointly faithful” from the definition of strong generators: that is, is a family of strong generators iff the functors are jointly conservative.
Proof. The functors preserve any limits that exist in .
Example. Let be the homotopy category of pointed connected CW complexes. Then Whitehead’s theorem asserts that the functors are jointly conservative, and these functors are hom functors , so they preserve all limits that exist in . However, they are not jointly faithful: for example, there are interesting homotopy classes of morphisms between Eilenberg-MacLane spaces whose homotopy groups live in distinct degrees, and for other examples see this MO question. It follows that fails to have equalizers.
What can we say about the converse? First, say that a morphism is a fake isomorphism (nonstandard) if it is epic and monic, but not an iso. For example, any continuous bijection which is not a homeomorphism is fake.
Proposition: Let be a faithful functor. If has no fake isos, then is conservative.
Proof. Since is faithful, it reflects epis and monos. If is an iso, then in particular it’s epic and monic, hence so is . By hypothesis, it follows that is an iso.
Corollary: If is a family of objects in a category with no fake isos, then is faithful iff it is strong.
In general, a category won’t have fake isos if its epis are well-behaved; for example, whenever epis are regular, a hypothesis we’ll use later as well. This holds in abelian categories but also in and ; however, it does not hold in , since for example is epic but not regular epic, and it does not hold in , since there epis are just surjections but regular epis are quotient maps.
Three definitions about colimits
The term “generator” can be interpreted as having something to do with generating a category under colimits. Here are three definitions along those lines. As in the previous section, we won’t make any explicit use of linearity.
Definition: Let be a cocomplete category. A family of objects is a family of
- naive generators (nonstandard) if every is a colimit of objects in ,
- presenting generators (nonstandard) if every is a coequalizer of a pair of morphisms between coproducts of objects in , and
- iterated generators (nonstandard) if every is an iterated colimit of objects in .
Mike Shulman uses the terms colimit-dense generator for naive generators and colimit generator for iterated generators respectively, but I find it hard to remember which is which (and in fact I mixed them up while writing this sentence). The idea behind the definition of presenting generators is that a coequalizer of the above form is a presentation by “generators” (the ) and “relations” (the ).
Example. is not naive, but it is presenting, iterated, strong, and faithful.
The five definitions we’ve introduced are totally ordered in strength as follows. As above, is a family of objects in a cocomplete category.
Theorem: Naive presenting iterated strong faithful.
Naive presenting: in a cocomplete category, every colimit can be computed as the coequalizer of a pair of morphisms between two coproducts.
Presenting iterated: every coequalizer of a pair of morphisms between coproducts is an iterated colimit (iterated twice: once for the coproducts, once for the coequalizers).
Iterated faithful: Suppose is iterated. We want to show that every admits some epi . Every satisfies this condition, and moreover it is closed under coproducts (since coproducts of epis are epis) and coequalizers (since coequalizer projections are epis, and epis are closed under composition), hence it is closed under colimits. So in fact every admits such an epi, and is faithful.
Iterated strong: Suppose is iterated. By the above, we know that the functors are jointly faithful, so it suffices to show that they are jointly conservative.
Say that a morphism is an -isomorphism if it induces an isomorphism
for all . If denotes the collection of all -isomorphisms, our goal is to show that consists precisely of the isomorphisms.
An object is -colocal if every -isomorphism induces an isomorphism . By hypothesis, every is -colocal. If is a colimit of -colocal objects and is an isomorphism, then
is an isomorphism since its components are. Hence the collection of -colocal objects is closed under colimits. Since is iterated, it follows that every is -colocal.
But this means that if is an -isomorphism, then is an isomorphism for all , and by the Yoneda lemma it follows that is an isomorphism. Hence is strong.
Strong faithful: by definition.
So far we’ve seen the following counterexamples to the converses of the above implications:
- shows that presenting naive.
- shows that faithful strong.
According to Mike Shulman, iterated strong under mild hypotheses (in addition to being cocomplete, it must have finite limits and satisfy a mild smallness condition), but we won’t use this. I expect that iterated presenting in general but don’t know a counterexample. Under additional hypotheses, we have the following.
Theorem: The implications presenting iterated strong faithful can be reversed under the following additional hypotheses:
- Faithful strong if has no fake isos (in particular if is abelian).
- Faithful presenting if every epi in is regular (in particular if is abelian).
- Iterated presenting if every is projective.
Faithful sometimes strong: proven previously.
Faithful sometimes presenting: suppose is faithful and let
be an epi. We would like to use this epi to exhibit as a coequalizer.
Now suppose that every epi is regular. Then is regular, hence is the coequalizer of some morphisms . By faithfulness, we can find another epi
and since coequalizers are unchanged by precomposing with an epi, we can replace with , and we conclude that
is a coequalizer diagram. Hence is presenting.
Iterated sometimes presenting: suppose is iterated. Let’s try to imitate the argument used in this post for discussing finite presentations in order to reduce any iterated colimit to a coequalizer of coproducts.
First, a colimit is a coequalizer of coproducts in the usual way. Since a coproduct of colimits can be expressed as a single colimit (take the coproduct of the corresponding diagrams), it suffices to show that we can express a coequalizer of colimits as a coequalizer of coproducts.
be morphisms, where , whose coequalizer we are trying to express as a coequalizer of coproducts. As before, admits an epi from a coproduct and coequalizers are unchanged by precomposing with an epi, so WLOG the above diagram has the form
Now suppose that every is projective. Then projective objects are closed under coproducts (assuming the axiom of choice: pick a lift for each object), so is projective, and the coequalizer projection is an epi, so lift to , and as before we find that we can reduce the above computation to the computation of a single coequalizer. Equivalently, WLOG the above diagram has the form
(although there may be more than there were before). Hence is presenting.
Corollary: If is a family of objects in an abelian category, then strong faithful. If is cocomplete, then presenting iterated strong faithful.
Now, in the previous post on tiny objects we proved various facts using naive generators, but an inspection of the proofs involved show that they go through for presenting generators (possibly even iterated generators, but we don’t need this; the point is that if a functor preserves colimits then it preserves iterated colimits), and using the above corollary we deduce the following.
Corollary (Freyd): If is a family of objects in a cocomplete abelian category, then the restricted Yoneda embedding is an equivalence of categories iff is a family of compact projective (faithful) generators.
Corollary (Gabriel): If is an object in a cocomplete abelian category, then
is an equivalence of categories iff is a compact projective (faithful) generator.
One last definition for abelian categories
Let be a linear category. In this setting the condition that is a family of (faithful) generators can be rephrased as the condition that the restricted Yoneda embedding reflects zero morphisms in the sense that if is a morphism in such that
is a zero morphism for all , then .
If is abelian, then a morphism is zero iff its image is zero, which suggests the following definition.
Definition: Let be a linear functor between linear categories. is weakly faithful (nonstandard) if it reflects zero objects: implies .
Definition: Let be a linear category. A family of objects is a family of weak generators if the restricted Yoneda embedding is weakly faithful.
More explicitly, this condition means that if is a nonzero object then there is some nonzero morphism .
Proposition: Let be a linear functor between linear categories. If is faithful, then it is weakly faithful. The converse holds if is exact and every morphism in has an epi-mono factorization (in particular if is abelian).
By an epi-mono factorization of a morphism we just mean a factorization where is monic and is epic. In an abelian category the image factorization accomplishes this. More generally, in a category with either 1) kernel pairs and coequalizers or 2) cokernel pairs and equalizers the regular coimage or regular image factorizations, respectively, accomplish this, so this is a very mild hypothesis.
Proof. : An object is a zero object iff the identity is a zero morphism. Since is faithful, it reflects zero morphisms.
: let be a morphism. By hypothesis, has an epi-mono factorization where fit into a diagram
Since is exact, it preserves epis and monos, and hence we get an induced epi-mono factorization .
If , then by cancelling on the left and cancelling on the right we conclude that , so . Since reflects zero objects, it follows that , so . Hence reflects zero morphisms as desired.
I expect that weak faithful in general but don’t know a counterexample.
Corollary: If is a family of objects in a cocomplete linear category, then naive presenting iterated strong faithful weak.
Corollary: If is a family of projective objects in an abelian category, then strong faithful weak. If is cocomplete, then presenting iterated strong faithful weak.
Corollary (Freyd): If is a family of objects in a cocomplete abelian category, then the restricted Yoneda embedding is an equivalence of categories iff is a family of compact projective weak generators.
Corollary (Gabriel): If is an object in a cocomplete abelian category, then
is an equivalence of categories iff is a compact projective weak generator.
More explicit Morita equivalences
Let be a ring. We now know that every Morita equivalence comes from a module which is a compact projective weak generator, where and the equivalence is given by
How explicit can we make this? is compact projective iff it is a finitely presented projective module, or equivalently a retract of a finite free module. This means that we can explicitly describe by writing down an idempotent
such that is the splitting of ; from here it follows that can be explicitly described as
This description exhibits as a “non-unital subring” of : the inclusion respects multiplication and addition but not the unit, and in fact the unit in is .
Since we already know that exhibits a Morita equivalence , we may assume WLOG that (by replacing with the Morita equivalent ), so that and . It remains to give an explicit criterion for when is a weak generator.
If , we want to determine when we can always find a nonzero map . The data of such a map is equivalent to the data of an element such that , and it is nonzero iff . So is a weak generator iff acts by a nonzero endomorphism on every module . The action of on any module factors through some quotient of (by a two-sided ideal), and conversely every such quotient appears this way, so the condition is precisely that is not contained in any proper two-sided ideal of .
Undoing the use of the Morita equivalence between and , we conclude the following.
Theorem: is a weak generator iff is a full idempotent, meaning that it is not contained in any proper two-sided ideal of , or equivalently, that .
More explicitly, the two-sided ideals of are precisely the submodules of regarded as an -bimodule in the usual way, or equivalently as an -module. Submodules are a categorical notion (corresponding to monomorphisms), so they are invariant under Morita equivalence. is Morita equivalent to , with the equivalence sending to regarded as an -module, or equivalently an -module in the usual way. And the two-sided ideals of are precisely the submodules of regarded as an -bimodule. Hence we conclude the following (which can also be proven without knowing anything about Morita equivalence):
Proposition: Every two-sided ideal of has the form where is a two-sided ideal of .
Corollary: is full iff for all proper two-sided ideals of .
The condition that is very strong: since , it follows by induction that for all , hence that . If is a commutative Noetherian domain and is a proper ideal, then by the Krull intersection theorem, , hence .
Corollary: If is a commutative Noetherian domain, then is full iff , so the rings Morita equivalent to are precisely the rings where is a nonzero finitely presented projective module.
Geometrically this has the following interpretation. If is commutative, then defines an algebraic vector bundle over the affine scheme , and the condition that is the condition that the restriction of to affine closed subschemes is nonzero. Idempotence implies that if the coefficients of vanish when restricted to , then they vanish to infinite order. The Krull intersection theorem implies that if this happens when is a Noetherian domain, then , so . (Compare to the statement that if a meromorphic function on a connected open subset of vanishes to infinite order at a point, then it vanishes identically.)
If we don’t assume that is a domain, then what can happen is that has a nontrivial decomposition as a product , so has a nontrivial decomposition as a coproduct , and a vector bundle over may be nonvanishing over one component but not the other. Algebraically this corresponds to the intersection being a nontrivial idempotent ideal in , such as one generated by a nontrivial idempotent.