The starting observation of Morita theory is that the abelian category of (right) modules over a (not necessarily commutative) ring does not uniquely determine , since for example we always have Morita equivalences of the form
Determining is equivalent to isolating the module (regarded as a module over via right multiplication), from which we can recover as its endomorphism ring. In some sense what this tells us is that cannot always be isolated in by a categorical property.
The next best thing we can try to do is to classify all of the rings such that by isolating the corresponding modules by some categorical property. The crucial property turns out to be that the hom functor
is faithful and preserves colimits. An object with this second property is called a tiny object, and in this post we’ll discuss how this condition behaves with an eye towards better understanding Morita equivalences. Along the way we’ll prove a theorem due to Gabriel characterizing categories of modules among abelian categories.
Every limit or colimit appearing in this post is over an essentially small diagram.
Every category appearing in this post will either be “ordinary” (-enriched) or “linear” (-enriched). We will be proving facts for both kinds of categories simultaneously because the proofs are almost identical.
If is an ordinary category then denotes the category of presheaves of sets on , while if is a linear category then denotes the category of presheaves of abelian groups on . Edit, 2/1/16: As Ingo Blechschmidt points out in the comments, this is a little ambiguous, so to be clear, by a presheaf of abelian groups on a linear category I mean a linear (-enriched) functor .
“Module” means “right module.” In particular, if is a ring regarded as a linear category with one object, then is the linear category of -modules.
Definition and examples
Let be a cocomplete category (possibly linear).
Definition: A tiny object is an object such that preserves colimits.
Warning. In the previous post about compact objects we didn’t talk about enrichments; for applications to module categories, this is because the forgetful functor preserves filtered colimits, and so in the definition of compactness we did not need to specify whether was regarded as taking values in sets or abelian groups. But the forgetful functor certainly does not preserve all colimits (e.g. it does not preserve coproducts), so here it’s crucial to talk about enrichments.
Example. If , then the one-object set is the only tiny object. To see this, if is a set, then in order for to preserve colimits it must in particular preserve finite coproducts, so the map
must be an isomorphism. The RHS can be identified with the set of subsets of , while the inclusion of the LHS can be identified with the inclusion of the empty subset and the subset consisting of all of . Hence if preserves finite coproducts, then must have exactly two subsets, the empty subset and , and this condition is satisfied precisely when .
Example. The motivating class of examples is that if is a category (possibly linear), then in its presheaf category , the representable presheaves are tiny by the Yoneda lemma, since colimits of presheaves are computed pointwise. These are the “free modules of rank ” on each . In particular, if is a one-object linear category with endomorphisms , we conclude that the free -module of rank is tiny in .
Example. Some categories have the property that the only cocontinuous functor is the constant functor with constant value the empty set (if has an initial object then this is the only possible constant value). Since a hom functor never has this property (it always has at least one nonempty value, on ), it follows that such a category has no tiny objects. Here is a class of examples and an example:
Subexample. Suppose has a zero object . Then is empty. Every object admits a map , but in the only set which admits a map to the empty set is the empty set, so is also empty. (In particular, abelian categories have no tiny objects when regarded as enriched over as opposed to enriched over .)
Subexample. Let be the category of commutative rings. Then is empty. The zero ring (which is not a zero object) is the coequalizer of the two projections . preserves coequalizers, so is also empty. But every ring admits a morphism , so admits a morphism to the empty set, hence is also empty. (While does not have a zero object, it does have the property that the terminal object is a colimit of copies of the initial object.)
Unlike the case of compact objects, which are closed under finite colimits because finite limits commute with filtered colimits (in both and ), at first it might seem like tiny objects have no closure properties under colimits. However, there are some (shapes for) colimits – the absolute colimits – which are preserved by any (enriched) functor whatsoever, and hence the corresponding absolute limits commute with arbitrary colimits.
Example. We saw in this blog post that zero objects are absolute for categories enriched over pointed sets, and finite coproducts / biproducts are absolute for linear categories.
Example. In both ordinary and linear categories, split idempotents are absolute. This is worth going into in some detail.
The facts of life about idempotents and retracts: Splitting idempotents is an absolute colimit, in both ordinary and linear categories. Explicitly:
- Let be an object and let be a retract of it, so that there are morphisms and such that . Let be the corresponding split idempotent. Then the retract , equipped with the map , is the coequalizer , and this coequalizer is preserved by any functor whatsoever. Dually, , together with the map , is the equalizer .
- Conversely, suppose is an idempotent in a category such that either the equalizer or the coequalizer of and exists. Then they both exist and are canonically isomorphic. The resulting object and the corresponding maps exhibit as a retract of , so and , and is a split idempotent. In particular, the equalizer and coequalizer of and are both preserved by any functor whatsoever.
This makes retracts, like zero objects and biproducts, another rare example of objects satisfying two universal properties, one with respect to maps in and one with respect to maps out.
Proof. We want to verify that , together with the map , satisfies the universal property of the above coequalizer. The universal property says that a map out of is the same thing as a map out of such that . So let’s verify this: given a map out of , the pullback is a map out of such that
In the other direction, given a map out of such that , the pullback is a map out of , and
Since everything we’ve just said involves only morphisms and equations between morphisms, it is preserved by applying an arbitrary functor as desired. Dualizing this argument gives the dual statement. (Working in the opposite category exchanges the roles of and , but it is still true that in the opposite category, is a retract of .)
Conversely, suppose is an idempotent and that the coequalizer of and exists; call it , and write for the corresponding map out of . By construction, a map corresponds, via pulling back along , to a map coequalizing . Since is idempotent, it is itself such a map; write for the corresponding map out of , which by construction satisfies . In particular,
Since is a coequalizer it is a regular epimorphism, and in particular an epimorphism, so it is right cancellative. Cancelling it gives
So equalizes . In fact exhibits as the equalizer of : if is a map into equalizing , so , then factors through since .
Since is an equalizer it is a regular monomorphism, and in particular a monomorphism, so it is left cancellative. Cancelling it from gives
and hence the maps exhibit as a retract of as desired.
Proposition: An absolute colimit of tiny objects is tiny.
Proof. Absolute colimits are preserved by all functors, and hence so are their corresponding absolute limits. Maps out of an absolute colimit is an absolute limit, and hence it commutes with colimits.
Explicitly, suppose is an absolute colimit of tiny objects and is an arbitrary colimit . Then by absoluteness, while by tininess, and hence
By absoluteness, is an absolute limit in the diagram category (since limits are computed pointwise), and hence is preserved by the colimit functor (or, for linear categories, ), so we can exchange the order of the limit and the colimit to get
But by a final application of absoluteness, and the conclusion follows.
Corollary: Retracts of tiny objects are tiny. In a linear category, finite biproducts of tiny objects are tiny.
Example. In , it follows that the retracts of finite free modules are tiny. These are precisely the finitely presented projective modules. More generally, if is a category (possibly linear), then it follows that the retracts (resp. retracts of finite biproducts) of the representable presheaves are tiny in the presheaf category .
Compact projective objects
“Tiny object” is a term that appears to only be used by category theorists. In nice linear situations, however, it is equivalent to a weaker (in general) condition, namely “compact projective object,” and this term is much more widely used.
If is tiny, then because preserves colimits, satisfies any other property which can be expressed in terms of preserving some restricted class of colimits. We’ll highlight three in particular:
- is connected: preserves finite coproducts. (This differs slightly from the nLab’s first definition but is equivalent to it in nice cases.)
- is compact: preserves filtered colimits.
- is projective: preserves epimorphisms (since it preserves pushouts).
The terminology of connectedness stems from the fact that in topological spaces, graphs, -sets and other categories where objects have a well-behaved decomposition into “connected components,” connectedness in the above sense is equivalent to connectedness in the usual sense. But this mostly won’t concern us here, although it is a nice way to think about why is the only tiny object in , since we showed above that it’s even the only connected object in .
What can we say about the converse?
Theorem: An object in a cocomplete category is tiny iff it is connected, compact, and preserves coequalizers. If is linear, then is tiny iff it is compact and preserves coequalizers, or equivalently cokernels. If is abelian, then is tiny iff it is compact and projective.
Proof. In a general cocomplete category (linear or otherwise), for a functor to preserve colimits it suffices that it preserve coproducts and coequalizers. An arbitrary coproduct is the filtered colimit of its constituent finite coproducts (the proof is similar to but simpler than the proof we presented in the previous post regarding arbitrary colimits), so for a functor to preserve colimits it suffices that it preserve finite coproducts, filtered colimits, and coequalizers.
In a linear category, finite coproducts / biproducts are absolute, so every object is connected, and coequalizers can all be computed as cokernels, so preserving one is equivalent to preserving the other. Finally, in a linear abelian category, preserves coequalizers iff is projective (as we showed earlier).
Since we also showed earlier that the compact objects in categories of modules are the finitely presented modules, we also conclude the following.
Corollary: In , the tiny objects are precisely the finitely presented projective modules, or equivalently the finitely generated projective modules, or equivalently the retracts of finite free modules.
Corollary: If are Morita equivalent rings, then is the endomorphism ring of a finitely presented projective -module.
Corollary: If is a ring such that every finitely presented projective -module is free, then the rings Morita equivalent to are precisely the rings .
This is nice to know, but it relies on a separate characterization of compact objects in which took some work to show. There is in fact a more direct and easier characterization of the tiny objects which we give below.
On the other hand, one reason to factor tininess into compactness and projectivity is that this whole story has a derived version in which, in some sense, every object becomes projective, so only compactness is left to worry about.
Warning. Some authors define a compact object in a cocomplete abelian category to be an object such that preserves coproducts. This is implied by compactness in the above sense but is not equivalent to it: for example, in categories of modules, it is satisfied by any finitely generated module (not necessarily finitely presented). It is not even equivalent to finitely generated for modules; see this MO discussion.
However, if is in addition assumed to be projective, so that preserves coequalizers, then preserves coproducts iff it preserves colimits iff is tiny, so with this definition of compact it is still true that “compact projective” is equivalent to “tiny.” And in a derived setting I think we can again pretend that every object is projective.
Recall that our strategy for characterizing the compact objects of a nice category was to write every object as a filtered colimit of some readily identifiable class of compact objects, from which it would follow that if is projective then the identity factors through a map to some member of this class, hence must be a retract of a member of this class. We can try this exact strategy for characterizing tiny rather than compact objects, and we get the following.
Suppose is a cocomplete category (possibly linear) and is an essentially small full subcategory of tiny objects of such that every object of is a colimit of objects in .
Example. Let be an essentially small category (resp. linear category) and let be its category of presheaves of sets (resp. abelian groups). If is an ordinary category, then by the Yoneda lemma every presheaf is a colimit of representables, and hence we can take to be the representables. If is a linear category, then the statement of the Yoneda lemma needs to be modified, but the correct linear Yoneda lemma implies (with some work) that every presheaf is a colimit of finite biproducts of representables, and hence we can take to be the finite biproducts of representables. This reduces to a familiar fact when has one object, namely that every -module is a colimit of finite free modules.
Unlike the hypotheses we used for studying compact objects, these hypotheses are very restrictive, and we will see later that presheaf categories are the only examples.
Theorem: With the above hypotheses, if is ordinary then the tiny objects of are precisely the retracts of the objects in , and if is linear then the tiny objects of are precisely the retracts of finite biproducts of the objects in .
Proof. Write a tiny object as a colimit where . By hypothesis, preserves colimits, so
If is an ordinary category, so that the above colimit is a colimit of sets, it follows that the identity factors through some , hence that is a retract of some .
If is a linear category, so that the above colimit is a colimit of abelian groups, it follows that the identity is a finite sum of morphisms factoring through some s, hence that it factors through some finite biproduct of the s, hence that is a retract of a finite biproduct of some s.
Corollary: If is an essentially small ordinary category, the tiny objects in are precisely the retracts of representable presheaves. If is a linear category, the tiny objects are precisely the retracts of finite biproducts of representable presheaves.
This gives a second and easier proof that the tiny objects in are precisely the retracts of finite free modules, or equivalently the finitely presented projective modules.
Example. Let be a monoid; we will use the same symbol to denote the corresponding one-object category. Then the category of presheaves is the category of -modules in set. The unique representable presheaf is the free -module on an element, or regarded as a module over itself. By the Yoneda lemma / Cauchy’s theorem, is the monoid of endomorphisms of (regarded as a module over itself), and hence idempotent endomorphisms of correspond to idempotent elements of . If is such an idempotent, then its retract in is the module generated by .
In particular, if has no nontrivial idempotents (e.g. if it is a group), then is the unique tiny -module.
Morita equivalences and Cauchy completion
We are now very close to characterizing Morita equivalences in the following sense.
Definition: Two essentially small categories (possibly linear) are Morita equivalent if .
When are rings thought of as linear categories with one object this reduces to the usual notion of Morita equivalence of rings, but this notion of Morita equivalence is substantially more general.
Example. One way to state the Dold-Kan correspondence is that the linearization of the simplex category (obtained by taking the free abelian group on its homsets; presheaves on this are simplicial abelian groups) is Morita equivalent to the linear category with objects labeled by the non-negative integers and morphisms the free abelian groups on generators satisfying . This is, by construction, the linear category such that presheaves on it are connective chain complexes.
If is a (Morita) equivalence, then it induces an equivalence on tiny presheaves. This motivates the following definition.
Definition: If is an essentially small category (possibly linear), then its Cauchy completion is the full subcategory of on the tiny presheaves.
For ordinary categories is also known as the Karoubi envelope, the Karoubi completion, or the idempotent completion of . As we more or less showed above, it is obtained from by formally splitting all idempotents, and can be constructed explicitly as follows:
- The objects of are pairs of an object and an idempotent .
- The morphisms are the morphisms such that . Equivalently, they are the morphisms of the form . Composition is as in .
The description of morphisms follows from the fact that the formal splitting of an idempotent has two universal properties: it is both the equalizer and the coequalizer of and . Note that this construction does not depend on being essentially small and makes sense for any category.
Example. Lawvere observed that the category of smooth manifolds and smooth maps is the idempotent completion of the category of open subsets of Euclidean spaces and smooth maps.
Example. A common step in the construction of categories of motives is to take the idempotent completion of a category of correspondences.
Similarly, as we more or less showed above, for linear categories is obtained from by first formally adjoining finite biproducts and then formally splitting all idempotents. Since we already know how to formally split idempotents it suffices to describe how to formally adjoin finite biproducts, which is done explicitly as follows:
- The objects are tuples of objects in .
- The morphisms are matrices of morphisms in . Composition is given by matrix multiplication as in the end of this blog post.
In general, when formally adjoining a colimit to a category, you know by definition what maps out of the colimit must be, but you have some freedom to decide what maps into the colimit are. In the special case of splitting idempotents and adjoining biproducts there is no such freedom: both of these have two universal properties, one with respect to maps out and one with respect to maps in, and so both are uniquely determined. More generally, there is no such freedom when adjoining absolute colimits because they are preserved by all functors, including hom functors .
Hence there is a unique way to adjoin an absolute colimit to a (possibly enriched) category, and in fact one way to describe the Cauchy completion (in a way that generalizes to other kinds of enriched categories) is that it is obtained by formally adjoining all absolute colimits. We can make this precise as follows.
Definition: A category (possibly linear) is Cauchy complete if it has all absolute colimits.
Note that we do not require that is essentially small, and in fact we will need to know that and are Cauchy complete (because they are cocomplete; for this statement should be interpreted in the linear sense). More generally, since we showed that tiny objects are closed under absolute colimits, the full subcategory of tiny objects in any cocomplete category is Cauchy complete.
In particular, the Cauchy completion as defined above is Cauchy complete. Let’s show that the Cauchy completion really deserves its name.
Theorem: Let be an essentially small (possibly linear) category and let be Cauchy complete (linear if is). Then the restriction functor from the category of (linear if is) functors to the category of functors is an equivalence. In particular, any functor extends essentially uniquely to a functor .
With the right construction of the Cauchy completion we can remove the hypothesis that is essentially small, and then the theorem above states precisely that Cauchy completion is the left (2-)adjoint to the inclusion (2-)functor from Cauchy complete categories to categories (possibly linear).
Proof. Using the explicit description of above, we’ll explicitly exhibit an inverse. If is a functor, its extension is uniquely determined by the fact that functors preserve absolute colimits: splits idempotents and, in the linear case, preserves finite biproducts. Similarly, if are functors and is a natural transformation, then its extension is uniquely determined by the fact that functors preserve absolute colimits naturally. All of the isomorphisms and compatibilities needed for this to be an inverse functor again follow from the fact that functors preserve absolute colimits naturally. (This is a lot more tedious to write out than it is to verify.)
Corollary: With the above hypotheses, is Cauchy complete iff is an equivalence iff the tiny objects in are precisely the representables iff all idempotents in split and, in the linear case, has finite biproducts.
Corollary: With the above hypotheses, the restriction functor is an equivalence; that is, is Morita equivalent to its Cauchy completion .
Corollary: Two essentially small categories (possibly linear) are Morita equivalent iff their Cauchy completions are equivalent.
This can be used to organize a proof of the Dold-Kan theorem; see this MO question for details.
Corollary: Let be two monoids with no nontrivial idempotents (e.g. groups). Then are Morita equivalent (meaning that their categories of modules in are equivalent) iff they are isomorphic.
This gives a proof, different from the proof given in this previous post, that a group is uniquely determined by the category of -sets.
The hypotheses on a cocomplete category we used above are very restrictive; the only examples we gave were presheaf categories. In fact, these hypotheses are so restrictive that they characterize presheaf categories.
Theorem: Let be a cocomplete (possibly linear) category and be an essentially small full subcategory of tiny objects in such that every object of is a colimit of objects in . Then the natural functor
is an equivalence.
In a sentence: presheaf categories are precisely the cocomplete categories with a family of tiny generators. (Here “family of generators” means the colimit condition above.) In the linear case this is due to Freyd.
Proof. We want to show that is fully faithful and essentially surjective. Let be two objects in , and write them as colimits of objects in . Since the are tiny,
Since every object of is tiny, the functor is cocontinuous, and it is automatically continuous, so applying this functor (and using the fact that is a full subcategory of and that the Yoneda embedding is full) preserves the above isomorphism. Hence is fully faithful.
Now let be a presheaf. By the Yoneda lemma, is a colimit of (in the linear case, finite biproducts of) representables. We can take the same colimit in , and because is cocontinuous, it gets preserved. Hence is essentially surjective, and the conclusion follows.
Corollary (Gabriel): Let be a cocomplete abelian category and be a compact projective object such that every object in is a colimit of finite biproducts of copies of . Then
is an equivalence.
Note that every equivalence from a linear category to a category of modules must have this form.
Proof. Apply the theorem to (rather than to ), then use the fact that is equivalent to the category of -modules because finite biproducts are absolute.
The hypothesis that is abelian is somewhat misleading: if “projective” is interpreted to mean “ preserves coequalizers” or “compact projective” is interpreted to mean tiny, then we only need to know that is a cocomplete linear category, and in particular we do not need to assume anything about limits in .
This theorem characterizes categories of modules among abelian categories by a categorical property, usually stated as “cocomplete and admits a compact projective generator,” and often misstated without the cocompleteness hypothesis. Note that “generator” in this statement means something slightly different than it did above; see this MO question.
This theorem also characterizes Morita equivalences between rings in the following sense: if is a ring, the rings Morita equivalent to are precisely the endomorphism rings of finitely presented projective modules which generate in the above sense.