Questions about Modules | The n-Category Café

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 CC a sufficiently nice braided monoidal category (semisimplicity needs to be assumed, for one), we have that every CC-module category is equivalent to a category of modules of an algebra object internal to CC.

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)→C−Mod \mathrm{Bim}(C) \to C-\mathrm{Mod} which sends algebras internal to CC to their categories of internal modules, sends bimodules internal to CC 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. \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 CC 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−Modk\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 [C,Ab][\mathbf{C}, \mathbf{Ab}] for some (small) Ab\mathbf{Ab}-enriched category C\mathbf{C}, where the square brackets denote the Ab\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 [C,Set][\mathbf{C}, \mathbf{Set}] for some small category C\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 [C,Set]→[D,Set][\mathbf{C}, \mathbf{Set}] \to [\mathbf{D}, \mathbf{Set}] is induced by a (C,D)(\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 MM-sets for some monoid MM. Comments as for the previous question.

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→Vect−Mod, \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 Vect\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 Bim C *→TopVect−Mod, \mathrm{Bim}_{C^*} \to \mathrm{TopVect}-\mathrm{Mod} \,, where on the left we have a notion of bimodules for C *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→Hilb−Mod. \mathrm{Bim}_{\mathrm{vN}} \to \mathrm{Hilb}-\mathrm{Mod} \,.

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

Aaron’s question I would reformulate like this:

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

Then what is the analog of the above questions for ΣBim(C)\Sigma \mathrm{Bim}(C)? It should be something about a morphism ΣBim(C)→(C−Mod)−Mod. \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 CC be a monoidal abelian category. Then a 2-vector space VV over CC is a module category over CC. In general, VV may or may not be equivalent to a category of modules over a given algebra (or algebroid, in fact) internal to CC.

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 nnLab 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 nnLab entry:

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

imf=lim(d→→d⊔ cd) 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 (∞,1)(\infty,1)-context the essential image of a morphism f:c→df : c \to d should be the homotopy limit holim(d→→d⊔ c hod)holim( d \stackrel{\to}{\to} d \sqcup^{ho}_c d ) of the homotopy fiber coproduct d⊔ c hodd \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

BU→∏c k∏ kB 2k−1U(1). \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 :-)