The n-Category Café

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.

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

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.

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

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. :)

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 $$\mathrm{Bim}(C) \to C-\mathrm{Mod}$$ 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 $$\mathrm{Bim}(C) \simeq C-\mathrm{Mod} \,.$$ 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).

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 $k\mathbf{-Mod}$, 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 $[\mathbf{C}, \mathbf{Ab}]$ for some (small) $\mathbf{Ab}$-enriched category $\mathbf{C}$, where the square brackets denote the $\mathbf{Ab}$-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 $[\mathbf{C}, \mathbf{Set}]$ for some small category $\mathbf{C}$. 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 $[\mathbf{C}, \mathbf{Set}] \to [\mathbf{D}, \mathbf{Set}]$ is induced by a $(\mathbf{C}, \mathbf{D})$-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.

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?

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

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

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 $$\mathrm{Bim}_{C^*} \to \mathrm{TopVect}-\mathrm{Mod} \,,$$ 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 $$\mathrm{Bim}_{\mathrm{vN}} \to \mathrm{Hilb}-\mathrm{Mod} \,.$$

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

Aaron’s question I would reformulate like this:

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

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

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.

I was googling the web trying to see who thought and taught what about the notion of essential image in a higher categorical context.

What I found is the $n$Lab entry [[essential image]] and this discussion here.

Probably there is more, and I need to look more closely. But I’ll post a question here anyway, related to the query box that I just added to the $n$Lab entry:

so there is the general definition of the image of a morphism $f : c \to d$ in a category with equalizers and pushouts as

$$im f = lim( d \stackrel{\to}{\to} d \sqcup_c d )$$

(see [[image]] for details of what I am talking about).

This definition has an obvious generalization in any higher context in whichh we have weak versions of colimits and limits.

I am in particular interested in the context of $\infty$-groupoids. Kan complexes if you like. or any other model category context.

In such an $(\infty,1)$-context the essential image of a morphism $f : c \to d$ should be the homotopy limit $holim( d \stackrel{\to}{\to} d \sqcup^{ho}_c d )$ of the homotopy fiber coproduct $d \sqcup^{ho}_c d$.

This must be a well known, well studied concept. Could someone just help me with collecting references. Or anything related.

To be very concrete, what I am actually interested in is the essential image of the product of all Chern classes

$$\mathbf{B} U \stackrel{\prod c_k}{\to} \prod_k \mathbf{B}^{2k-1} U(1) \,.$$

Any comment is appreciated.

I’ve been reading this thread on the hunt for a related question.

Is it the case that any abelian variety of algebras (I sort of want to apply bracketing here: I mean a variety of algebras which is an abelian category) is in fact a category of S-modules for some ring S?

Or another way of asking: if I have a category of R-modules and a Birkhoff subcategory of it (Birkhoff subcategory = full reflective subcategory closed under subobjects and regular quotients; in the context of varieties of algebras, it just means subvariety), is this also a category of S-modules for some ring S? (Rather than just a subcategory of such a thing.)

I guess given the answers with the small/compact projective generators, I should try to work out

a) Does a Birkhoff subcategory have all small coproducts? I think the answer is yes: it inherits them from R-Mod because the reflector, which is a left adjoint, so preserves colimits. (So just any reflective subcategory inherits colimits that exist in the ambient category.)

b) Does FR become a compact projective generator? (Since we know that R is a compact projective generator in R-Mod.)

I guess I’ll have to work this out, but maybe someone already knows and can tell me :-)