Questions about Modules

August 5, 2007

Posted by John Baez

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Here’s another question that’s been bugging me a long time. I had so much luck with the last one that I’m feeling slightly optimistic. We’ll see.

Question: Which categories are categories of modules for some ring? Which functors between these are given by tensoring with some bimodule? And which natural transformations between those are given by tensoring with some bimodule morphism?

I can make this a bit more precise…

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One of the first weak 2-categories you’re likely to meet is called $Bimod$ . It has:

rings for objects,
bimodules for morphisms,
bimodule homomorphisms for 2-morphisms.

There’s a weak 2-functor

$Mod : Bimod \to AbCat$

where $AbCat$ is the strict 2-category with:

abelian categories for objects,
right exact functors for morphisms,
natural transformations for 2-morphisms.

It works like this:

given a ring $R$ , $Mod (R)$ is its abelian category of modules,
given a bimodule $M : R \to S$ (that is, an $R$ - $S$ bimodule), $Mod (M) : Mod (R) \to Mod (S)$ is the right exact functor ‘tensoring with $M$ ’,
given an $R$ - $S$ bimodule homomorphism $f : M \Rightarrow N$ , $Mod (f) : Mod (M) \Rightarrow Mod (N)$ is the natural transformation ‘tensoring with $f$ ’.

Here’s my question:

Question: What is the ‘essential image’ of the 2-functor $Mod$ ?

In other words:

which abelian categories are equivalent to a category of modules of some ring?
which right exact functors $Φ : Mod (R) \to Mod (S)$ are naturally isomorphic to a functor given by tensoring with some $R$ - $S$ bimodule?
which natural transformations $ϕ : Mod (M) \Rightarrow Mod (N)$ come from a bimodule homomorphism $f : M \Rightarrow N$ ?

Any nice answers to these rather open-ended questions would be interesting to me.

I’d be even happier if we could work over an arbitrary commutative ground ring $k$ . We have a weak 2-category ${Bimod}_{k}$ with:

algebras over $k$ for objects,
bimodules of such algebras for morphisms,
bimodule homomorphisms for 2-morphisms.

There’s a weak 2-functor

${Mod}_{k} : {Bimod}_{k} \to {AbCat}_{k}$

where $AbCat$ is the strict 2-category with:

$k$ -linear abelian categories for objects,
$k$ -linear right exact functors for morphisms,
natural transformations for 2-morphisms.

It works just the same way. So again:

Question: What is the essential image of the 2-functor ${Mod}_{k}$ ?

It’s possible that this question would have a much nicer answer if we used comodules of coalgebras instead of modules of algebras. If so, I’ll be happy to switch to comodules! I’ll follow the tao where it leads me.

By the way: my question is slightly related to Allen Knutson’s question on Schur functors, and my conjectured answer. And, if we take $k = ℂ$ , it’s somewhat related to Urs Schreiber’s thoughts on $Vect$ -modules as fibers of generalized 2-vector bundles suitable for elliptic cohomology. So, it keeps coming up. But, it just seems like a good question to know the answer to!

Posted at August 5, 2007 1:30 PM UTC

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Re: Questions about Modules

The question is well-known, see e.g. Victor Ginzburg’s “Lectures on Noncommutative Geometry”, math.AG/0506603. Briefly, an abelian category is equivalent to the category of left modules over a ring if and only if it admits arbitrary direct sums and has a compact projective generator.

Posted by: Oleksandr Manzyuk on August 5, 2007 3:56 PM | Permalink | Reply to this

Re: Questions about Modules

I think it’s all right exact functors that preserve colimits. (This is “Watt’s theorem”, IIRC.)

Posted by: Walt on August 5, 2007 4:17 PM | Permalink | Reply to this

Re: Questions about Modules

For others of John’s questions I believe the magic phrase is Morita theory. It is interesting to note that there is a form of Mortia theory giving descriptions of equivalences between categories of torsors. It is briefly described in Breen’s article on Bitorsors but he refers to Giraud for a proof.

Posted by: Tim Porter on August 5, 2007 9:56 PM | Permalink | Reply to this

Re: Questions about Modules

This is the Yoneda lemma if we only ask about 1-categories and 1-functors, right?

Posted by: Mikael Johansson on August 5, 2007 10:17 PM | Permalink | Reply to this

Re: Questions about Modules

It seems to me that it’s kind of against the spirit of the question to say that the essential image consists of those categories with a compact projective generator. I mean, it’s not that much better than saying that the essential image consists of the abelian categories such that [repreat the definition of essential image here].

In other words, it seems to me that if you really want to follow the Tao of Morita theory, then picking a projective generator is bad. I have in mind the following analogies:

Tannakian category : algebraic group
Topos : site covered by a discrete site
Morita category : ring

where Morita category is the kind of category we’re looking for. On the left-hand side, you have categories defined by certain nice, *internal* properties. Then if you can find a fiber functor / enough points / a compact projective generator, then you can represent your category in a concrete way. But you don’t want to force yourself to have to do this.

At least that’s what I would have meant if I had asked the question. :)

Posted by: James on August 6, 2007 8:07 AM | Permalink | Reply to this

Re: Questions about Modules

OK, I see your point. Indeed, it would be nice to have an intrinsic characterization of module categories among abelian categories. I’m curious if somebody has ever looked at the problem from this perspective.

Posted by: Oleksandr Manzyuk on August 6, 2007 9:41 AM | Permalink | Reply to this

Re: Questions about Modules

It seems to me that it’s kind of against the spirit of the question to say…

I’m sufficiently ignorant, and desperate for knowledge, that all answers to my question are helpful.

You’re right that I’d prefer “intrinsic” properties of categories, functors and natural transformations as answers to my questions. I can guess how a “compact projective generator” is just a slightly disguised way of talking about a fiber functor (even though I don’t know what “compact” means here), which is not very “intrinsic”.

But still, I hadn’t known the characterization Oleksandr Manzyuk states, nor had I heard of “Watt’s theorem”, so I’m very grateful to Oleksandr and Walt for their help, and the reference to Ginzburg’s paper.

Of course, I’d also love to hear other answers.

Posted by: John Baez on August 6, 2007 11:52 AM | Permalink | Reply to this

Re: Questions about Modules

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i’ve been wanting to know the answer to these questions too, for ages. i’d also like to know how Mitchell’s embedding theorem fits into this picture.

Posted by: interested_user on August 6, 2007 12:22 PM | Permalink | Reply to this

Re: Questions about Modules

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James, can you explain the “Topos : site covered by a discrete site” part of that analogy? In particular, what is a discrete site? And what does it mean for one site to cover another?

My guess is that the story goes something like this. Theorem: a category is a Grothendieck topos iff it is equivalent to a subtopos of some presheaf topos ${Set}^{C^{op}}$ . (That bit’s not a guess.) Perhaps a site is “discrete” if the only covering sieves are maximal, so that a sheaf on it is merely a presheaf. In that case, every topos of sheaves on a site (i.e. every Grothendieck topos) embeds in some topos of sheaves on a discrete site.

Taking sheaves turns sites into toposes. (I’m guessing you’re using “topos” to mean “Grothendieck topos”, and I’ll do the same.) Special case: taking sheaves turns discrete sites into presheaf toposes. The fact that every topos embeds into a presheaf topos is perhaps a consequence of some statement like “every site is covered by a discrete site”. Obviously, some duality is involved.

But that’s all a guess.

Posted by: Tom Leinster on August 6, 2007 4:53 PM | Permalink | Reply to this

Re: Questions about Modules

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I was a bit sloppy with my table of analogies. That’s probably what confused you, but just in case, I’ll explain what I meant in the topos line. (And don’t know if your guess is right because now I can’t understand what you wrote.)

Truth be told, I was kinda shooting from the hip on that point, but I think it’s OK. By discrete site, I just meant a site $S$ which comes from a discrete topological space. That’s the same as just some big product of SET. And it’s probably equivalent to every presheaf being a sheaf. By a covering of a topos $T$ by $S$ , I meant a geometric morphism $f : \tilde{S} \to T$ which has the property that $f^{*}$ is faithful. Here, $\tilde{S}$ means the topos of sheaves on $S$ . (I think this is equivalent to giving a site map $S \to T'$ , where $T'$ is the site underlying $T$ .) It then follows that $f^{*}$ is comonadic, so $T$ can be represented as the category of sheaves on $S$ equipped with an action of comonad. So, like the Tannakian business, our category is the category of representations of something (= a comonad) on a category we think of as being understandable (= products of sets).

The flaw in my table is that while not all toposes have enough points, all Tannakian categories do have fiber functors, by definition (now that I’ve looked it up). And of course one of the categories in John’s essential image would have to be a fiber functor.

So, now that you’ve made me actually think through things a little more carefully, I see that there are many ways of rigidifying one of these categories. For instance, for toposes, you can consider:

1. Toposes

2. Toposes coverable by a something in your favorite class $C$ of toposes (e.g., topological spaces or discrete topological spaces)

2’. Same as 2, but with the covering map part of the data, rather than its existence being a property

3 and 3’. Same as 2 and 2’ but with $C$ very trivial (e.g., just the single point topos)

And then the Tannakian picture could be this:

1. Tensor categories ( $k$ -linear for a fixed field $k$ )

2. Tannakian categories (=tensor categories which admit a fiber functor to vector bundles over an algebraic space)

2’. Same as 2 but with a fixed fiber functor

3 and 3’. Same as 2 and 2’ but where the fiber functor takes values in vector spaces (=vector bundles over the point)

I think 3’ is what people usually call a neutral Tannakian category.

So, you might imagine something similar for Morita theory. I would say that an abelian category with a compact projective generator, which is the same as a fiber functor to abelian groups, would be the analogue of 3’. I think that the many-object version you mentioned in your post below would be something along the lines of 2’.

But for me the most interesting thing would be the analogue of 1. Could this just be an abelian category? If so, I would hope that the there would be some theorem saying that a large class of things of type 1 are actually of type 2. For example, Deligne proved that all coherent toposes have enough points. And much later he proved (someone should give that guy a medal) that over a field $k$ of characteristic 0, any $k$ -linear tensor category is of type 2.

If something like this is true, it then seems a bit unlikely that the Morita version of 1 is an abelian category. This is because the category of quasi-coherent sheaves on any scheme is an abelian category, but surely they’re rarely the same as categories of $R$ -modules.

Maybe the right thing is the category of $R$ -modules, where $R$ is a ring in a topos?

Posted by: James on August 7, 2007 12:31 PM | Permalink | Reply to this

Re: Questions about Modules

James wrote that “the category of quasi-coherent sheaves on any scheme is an abelian category, but surely they’re rarely the same as categories of R-modules”.
(One day I’ll learn how to honestly quote posts!)

I was (uncharacteristically) trying to avoid saying the word derived everywhere, but failed: it might be useful to note here that
for any (quasicompact, quasiseparated) scheme, its DERIVED categories of quasicoherent sheaves is equivalent to modules over a differential graded algebra. This seems to be one indication that the derived version of Morita theory is much better behaved. One key point I think is that in usual Morita theory we seek projective generators, which might be harder to come by than just plain generators, which in the dg context suffice.
I’ll avoid making [more of] a fool of myself with getting all the technical hypotheses wrong but refer to
the awesome Inventiones paper by Bertrand Toen on Derived Morita Theory, arXiv:math/0408337, or Keller’s great ICM address, arXiv:math/0601185, where
these things are described.

In particular I think John’s questions have very nice answers in this context: Toen proves a statement that roughly speaking says all functors (with some continuity assumption) between two differential graded categories are given by bimodules. Since reasonable
dg categories always tend to be module categories over a dga [observe me avoiding all technical pitfalls by vagueness!]
this says as I understand it roughly that
the world of dg categories is equivalent to the world of dgas with bimodules
(and this seems to be a basis of noncommutative algebraic geometry).

Posted by: David Ben-Zvi on August 7, 2007 3:59 PM | Permalink | Reply to this

Re: Questions about Modules

Addendum: The theorem the schemes are
affine in the derived sense (ie that
quasicoherent sheaves on a [quasicompact quasiseparated, ie “any”] scheme are derived equivalent to a dg algebra) is due to Bondal and van den Bergh, arXiv:math/0204218.

The subject started I think from Beilinson’s realization way back in 1978 that P^n was derived equivalent to a finite dimensional algebra (a quiver with relations): you just take the object
E which is the sum of O,O(1),O(2),…O(n)
and it generates everything, so the
whole category is derived equivalent to modules over its endomorphism algebra.

That also gives a concrete realization of the above theorem for quasiprojective varieties X in P^N: just take the (derived) restriction of the generator E for P^N and you get a derived generator for X, ie sheaves on X are equivalent to modules over the self-Exts dga of E on X.
This is put to great use in the homological mirror symmetry world of Kontsevich, Seidel etc.

I would be remiss not to mention that many of these structural (Morita and generation) results for dg categories have analogues (some earlier) in the much harder world of stable categories - things like categories of spectra (which rationally are the same as dg categories) – I don’t know the literature but Keller’s ICM gives a nice overview; I think a lot of the results are due to Schwede and Shipley, see the review arXiv:math/0310146.

Posted by: David Ben-Zvi on August 7, 2007 4:18 PM | Permalink | Reply to this

Re: Questions about Modules

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Thanks James. What had confused me was the following. I know what it means for one topos to cover another, but you were talking about one site covering another.

Typically we view maps of sites as going in the opposite direction than maps of toposes. For instance, a continuous map $X \to Y$ of topological spaces induces a morphism $Open (Y) \to Open (X)$ of the resulting sites and a geometric morphism $Sh (X) \to Sh (Y)$ of the resulting toposes. So if there’s a notion of one site $D$ covering another site $C$ , you’d expect it to correspond to the topos $Sh (C)$ embedding in the topos $Sh (D)$ .

Posted by: Tom Leinster on August 7, 2007 10:00 PM | Permalink | Reply to this

Re: Questions about Modules

The analogy you probably want Grothedieck category : ring. Grothendieck categories have all the properties of categories of modules, but lack projective generators (or projective modules, in general).

Posted by: Walt on August 7, 2007 5:47 PM | Permalink | Reply to this

Re: Questions about Modules

Thanks. Grothendieck categories seem like a good start. There’s also something called the Gabriel topology associated to an abelian category, and it looks like that it’s supposed to be the ringed space whose module category tries to be the original abelian category. But I haven’t been able to find any references that I can read. So I don’t really have anything useful to say beyond bringing them up.

Posted by: James on August 9, 2007 1:28 PM | Permalink | Reply to this

Re: Questions about Modules

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What got me started on all this originally was Ostrik’s theorem that for $C$ a sufficiently nice braided monoidal category (semisimplicity needs to be assumed, for one), we have that every $C$ -module category is equivalent to a category of modules of an algebra object internal to $C$ .

At the last CFT workshop in Oberwolfach I asked Viktor Ostrik if he had thought about whether and how this result extends to a theorem saying something about the resulting map $Bim (C) \to C - Mod$ which sends algebras internal to $C$ to their categories of internal modules, sends bimodules internal to $C$ to the functors between these categories obtained the way you indicated, etc.

His reply was (if I remember correctly) that he had not written this anything on this extended statement yet, but that he thought that this does in fact yield an equivalence $Bim (C) ≃ C - Mod .$ But don’t take my word for it. I might be misremembering exactly what he said. And keep in mind that lots of rather strong assumptions on $C$ enter here (which however do happen to have interesting examples in rational CFT).

Posted by: Urs Schreiber on August 6, 2007 2:44 PM | Permalink | Reply to this

Re: Questions about Modules

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This is a reply along the lines of the following old joke. Lost person accosts local and asks “what’s the best way to get to Riverside?”; local replies “well, I wouldn’t start from here”. In other words, it may not be helpful. Regardless:

Your question involves rings, that is, one-object Ab-enriched categories. You could ask three different questions, by changing “one-object” to “many-object” and/or “Ab-enriched” to “Set-enriched”. (Of course, you can consider enrichment in other categories still, including $k - Mod$ , as you mention.)

The three other questions (to which I do not know the answers) are as follows.

Many-object, Ab-enriched: here you’re asking which abelian categories are of the form $[C, Ab]$ for some (small) $Ab$ -enriched category $C$ , where the square brackets denote the $Ab$ -enriched functor category. And, of course, you’re asking the accompanying questions for functors and transformations.

Many-object, Set-enriched: here you’re asking which categories are presheaf categories, i.e. of the form $[C, Set]$ for some small category $C$ . While I don’t know the answer, I’m almost certain that competent topos theorists do. I do know the answer for the accompanying question about functors: a functor $[C, Set] \to [D, Set]$ is induced by a $(C, D)$ -module iff it preserves colimits (and in that case, it’s induced by an essentially unique such module). All transformations are induced by module homomorphisms; more precisely, the 2-functor analogous to the one you defined is locally full and faithful.

One-object, Set-enriched: here you’re asking which categories arise as the category of $M$ -sets for some monoid $M$ . Comments as for the previous question.

Posted by: Tom Leinster on August 6, 2007 5:23 PM | Permalink | Reply to this

Re: Questions about Modules

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Here you’re asking which categories are presheaf categories, i.e. of the form [C,Set] for some small category C.

(Is there an Australian category theorist in the house?)

I hope someone more expert than me will weigh in at some point, but I sort of wondered about this myself over here. I should have just consulted the reference I gave:

R. Rosebrugh and R. Wood, An adjoint characterization of the category of sets (1994).

The answer is a bit on the formal side, but it might be a reasonable start. It’s also a pure piece of enriched category theory (and so it applies to Tom’s other question, on characterizing additive categories of the form ${Ab}^{C}$ ), although I’ll write this comment just for the case of Set-enriched (i.e., locally small) categories.

To set the stage, recall that Giraud’s theorem characterizes Grothendieck toposes by means of exactness and generating set conditions. An elegant characterization theorem in this spirit is given in

Ross Street, Notions of Topos, Bull. Austral. Math. Soc. 23 (1981), 199-208.

It says that a Grothendieck topos $E$ is a lex total category, i.e., the Yoneda embedding

Y : E \to {Set}^{E^{op}}

admits a left adjoint

M : {Set}^{E^{op}} \to E

which preserves finite limits. And conversely, a lex total (locally small) category $E$ satisfying a mild size condition is a Grothendieck topos.

(The size condition is simply this: thinking of $Set$ as a ‘set’ in an ambient Grothendieck universe $U$ , the class of isomorphism classes of objects in $E$ is a ‘set’ of cardinality no greater than that of $Set$ .)

Lex totality is thus a compressed way of expressing all of the exactness properties of a Grothendieck topos. (The analogue for posets or 2-enriched categories is much better known: a poset $P$ whose “Dedekind-Yoneda” embedding

P \to 2^{P^{op}} = Down (P)

admits a finite-meet-preserving left adjoint $\sup : 2^{P^{op}} \to P$ is the same thing as a locale [or frame].)

Rosebrugh and Wood give a characterization of presheaf toposes in a similar vein, in terms of strengthened exactness conditions and a size condition.

In the first place, as noted here, presheaf categories $E = {Set}^{D^{op}}$ are totally distributive: their Yoneda embeddings have left adjoints which in turn have left adjoints:

L ⊣ M ⊣ Y : E \to {Set}^{E^{op}}

(In the world of ordered sets, 2-Cat, total distributivity is a constructive version of complete distributivity, where arbitrary meets distribute over arbitrary joins.) Conversely, suppose given a totally distributive category $E$ . There is a canonical map

λ : L \to Y : E \to {Set}^{E^{op}}

mated to the inverse of the counit, $ε^{- 1} : 1 \to M Y$ . Let $i : C \to E$ be the inverter of $λ$ , i.e., let $C$ be the full subcategory of objects $c$ of $E$ for which $λ_{c}$ is an isomorphism. For $E = {Set}^{D^{op}}$ , this $C$ is the Cauchy completion of $D$ .

Then, if $i$ is dense (i.e., if the functor $E \to {Set}^{C^{op}}$ sending $e$ to $E (i (-), e)$ is full and faithful), and under the condition that $C$ is essentially small (so that ${Set}^{i^{op}}$ admits a left adjoint $\exists_{i}$ ), $E$ is a presheaf topos.

Proof: The functor $E \to {Set}^{C^{op}} : e \mapsto E (i (-), e)$ has a left adjoint

{Set}^{C^{op}} \overset{\exists_{i}}{\to} {Set}^{E^{op}} \overset{M}{\to} E

which sends a weight $X : C^{op} \to Set$ to the weighted colimit $X \otimes_{C} i$ . We show the unit and counit of this adjunction are isomorphisms. The counit is an isomorphism because the right adjoint is fully faithful (density of $i$ ). The inverse to the unit is provided by a chain of isomorphisms

E (i (-), M \exists_{i} (X)) ≅ {Set}^{E^{op}} (L i (-), \exists_{i} X) ≅ {Set}^{E^{op}} (Y i (-), \exists_{i} X) ≅ ({Set}^{i^{op}} \circ \exists_{i}) (X) ≅ X

where we have used, in turn, the adjunction $L ⊣ M$ , the fact that $i$ is the inverter of $λ : L \to Y$ , the Yoneda lemma, and full faithfullness of $i$ . QED

For applications, it would be nice to have a more concrete understanding of what, e.g., density of $i$ really means, but the main point is that total distributivity is the crucial concept.

Posted by: Todd Trimble on August 8, 2007 5:49 PM | Permalink | Reply to this

Re: Questions about Modules

I have an extension of John’s question. Tensoring over the ground ring makes Bim_k into a (weak) monoidal 2-category. Does anyone have any idea when the 2-functor Mod: Bimod –> AbCat can be extended to a monoidal 2-functor?

Posted by: Aaron Lauda on August 8, 2007 11:50 AM | Permalink | Reply to this

Re: Questions about Modules

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Maybe to add to Aaron’s question:

to me, it seems that the following are the important issues to be understood properly:

What can be said about the morphism

$Bim \to Vect - Mod,$ where on the right we have the 2-category of categories which are modules over the 2-monoid $Vect$ .

To what degree is this monic, for instance?

Or, possibly, we actually want this question to be posed in a context with a little more extra structure around:

What can be said about the canical 2-functor ${Bim}_{C^{*}} \to TopVect - Mod,$ where on the left we have a notion of bimodules for $C^{*}$ -algebras, and on the right something like module categories over the category of topological vector spaces.

And so on. For instance for von-Neumann algebras ${Bim}_{vN} \to Hilb - Mod .$

More generally, for any abelian monoidal category $C$ we have a bicategory $Bim (C)$ of algebras and bimodules internal to $C$ , and we would like to understand the properties of the canonical morphism $Bim (C) \to C - Mod .$

Aaron’s question I would reformulate like this:

whenever $C$ is not just monoidal, but braided monoidal, $Bim (C)$ is monoidal and we get a 3-category $Σ Bim (C)$ where composition along the single object is the monoidal product in $Bim (C)$ .

Then what is the analog of the above questions for $Σ Bim (C)$ ? It should be something about a morphism $Σ Bim (C) \to (C - Mod) - Mod .$

Posted by: Urs Schreiber on August 8, 2007 1:26 PM | Permalink | Reply to this

Re: Questions about Modules

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Since the question came up again in another discussion, the following maybe deserves to be said again:

Let $C$ be a monoidal abelian category. Then a 2-vector space $V$ over $C$ is a module category over $C$ . In general, $V$ may or may not be equivalent to a category of modules over a given algebra (or algebroid, in fact) internal to $C$ .

But if it is, we may regard the choice of equivalence as a choice of 2-basis.

Hence categories of modules are like those 2-vector spaces which admit a basis.

Posted by: Urs Schreiber on August 13, 2007 2:07 PM | Permalink | Reply to this

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