### Questions about Modules

#### Posted by John Baez

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…

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 $\Phi: Mod(R) \to Mod(S)$ are naturally isomorphic to a functor given by tensoring with some $R$-$S$ bimodule?
- which natural transformations $\phi : Mod(M) \Rightarrow Mod(N)$ 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 $k = \mathbb{C}$, 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!

## 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.