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Background

Let $R$ be a (as always commutative) ring. Consider the $2$-category $\text{Cat}_{c\otimes}(R)$ of cocomplete $R$-linear tensor categories. There are various reasons why $\text{Cat}_{c\otimes}(R)$ can be seen as a categorified version of the category of $R$-algebras. For example, coproducts are categorified sums, tensor products are categorified products, and the imposed cocontinuity of the tensor product can be seen as a sort of distributive law. There is a $2$-functor $\text{Alg}(R) \to \text{Cat}_{c\otimes}(R), A \mapsto Mod(A)$ ("categorification"), which is $2$-left adjoint to $\text{Cat}_{c\otimes}(R) \to Alg(R), C \mapsto \text{End}(1_C)$ ("decategorification"). See also Lurie's article Tannaka duality for geometric stacks for evidence that that this category is important in algebraic geometry: The category of geometric stacks over $R$ embeds via quasi-coherent sheaves fully faithful into $\text{Cat}_{c\otimes}(R)$ (if we restrict to so called tame tensor functors). See also this entry in Todd Trimble's blog.

Question

Is $\text{Cat}_{c\otimes}(R)$ a locally presentable $2$-category?

Note that I don't want to consider it as a $(2,1)$-category here, but you may assume this if it is really necessary. To avoid set-theoretic difficulties in the following, perhaps we should restrict to categories with $\kappa$-small colimits for a fixed regular cardinal $\kappa$. Then $\text{Cat}_{c\otimes}(R)$ is probably $2$-cocomplete, see Mike Shulman's answer here. It already seems to be hard to describe colimits explicitly (see here).

There is a free cocomplete $R$-linear tensor category on one object, explicitly given by $\text{Mod}(R)^{\mathbb{N}}$ with a convolution tensor product. Imagine this as the categorified ring of power series over $R$. Thus $Hom(\text{Mod}(R)^{\mathbb{N}},-)$ is identified with the forgetful $2$-functor $\text{Cat}_{c\otimes}(R) \to \text{Cat}$, which is, however, not faithful since the data of a cocont. tensor functor does not only consist of a funcor, there is also the natural isomorphism expressing the compatibility with the tensor structure. Thus $\text{Mod}(R)^{\mathbb{N}}$ is not a generator. Remark that in the decategorified setting, the free $R$-algebra on one object, namely the polynomial algebra $R[x]$ is a generator of $\text{Alg}(R)$.

But maybe $\text{Cat}_{c\otimes}(R)$ is too big to be generated by a set of $\lambda$-presentable objects? If this is not the case, what is a reasonable full subcategory which contains the categories of quasi-coherent sheaves and is locally presentable as a $2$-category?

EDIT: Motivated by Jacob Lurie's answer, I would like to change the question to the following: Let $\lambda$ be a regular cardinal and $\text{Cat}_{c\otimes}^\lambda(R)$ denote the $2$-category of locally $\lambda$-presentable $R$-linear tensor categories (of course the definition of this includes that the $\lambda$-presentable objects form a submonoid with respect to $\otimes$). Is it a locally presentable $2$-category?

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Martin, you're asking some great questions these days, and insofar as we (and others) are talking with James Dolan, I'd really like to make more time to think harder about them. I have a couple of drafts of answers which I wanted to ripen a little further, but this comment is just to say that I for one am not ignoring your questions! It might not be a bad idea to discuss some of this offline... –  Todd Trimble Jun 3 '11 at 14:43
    
(What do I mean, "these days"? Your questions are always good! :-) –  Todd Trimble Jun 3 '11 at 14:45
    
+1. $\operatorname{Cat}_{c\otimes}(R)$ ought to be presentable, regardless of whether it is. In recent work with Alex Chirvasitu, we convinced ourselves that (as we defined it) $\operatorname{Cat}_{c\otimes}(R)$ is cocomplete, but since we didn't need it, we didn't write up the proof. Note that you must assume that $R$ is commutative, or otherwise that it comes equipped with some extra structure; else $\operatorname{Mod}(R)$ and the rest are not monoidal categories (otherwise I'm not sure what it would mean to be enriched over $R$). –  Theo Johnson-Freyd Jun 6 '11 at 1:30

1 Answer 1

Your 2-category can be described as commutative algebras over "R-mod" in the setting of cocomplete categories (at least this is one meaning of the word "R-linear", which does not imply that the underlying category is abelian. If you want to stick to abelian categories the answer will be the same but require some more words.)

So a more basic question is: is the collection of cocomplete categories (maybe with some adjectives attached) locally presentable? If you fix a regular cardinal $\kappa$, require all categories to be generated by a set of $\kappa$-compact objects, and all functors to preserve $\kappa$-compact objects, then the answer is yes. For example, if $\kappa$ is uncountable, then this 2-category is equivalent (via the functor which formally adjoins $\kappa$-filtered colimits) to the 2-category of small categories which admit $\kappa$-small colimits, and functors which preserve $\kappa$-small colimits (this 2-category is "algebraic" in nature, albeit with respect to operations of (fixed) infinite arity).

If you don't restrict your functors to preserve $\kappa$-compact objects, then it's not reasonable for the underlying $(2,1)$-category to be locally presentable because it is not even locally small. Your post contains an example: there's a free $R$-linear tensor category on one generator, and the category of $R$-linear tensor functors from that to some target category is equivalent to the target category, which is typically not small. (There is probably some reasonable way to salvage the situation if you take advantage of the full 2-category structure: the morphism categories in your example will not be small but they are nevertheless accessible.)

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Thank you for this answer! I would like to comment each paragraph: 1) This is a nice description, which also fits into the categorified picture: Imagine (finitely) cocomplete categories as categorified abelian groups. By $R$-linear, I mean enriched in $Mod(R)$. 2) Where can I find a proof that this simplified category is locally $2$-presentable? I understand that it is somehow $2$-algebraic, but I even have not digested yet the proof that it is $2$-cocomplete which Mike Shulman mentioned. I haven't found it in your HTT. –  Martin Brandenburg Jun 3 '11 at 22:31
    
3) Size issues are complicated here. The set of objects of $Cat_{c\otimes}$ is not a subset of our given universe $\mathcal{U}$, so why should be expect the hom-sets to be elements of $\mathcal{U}$? Perhaps we should jump to a bigger universe and ask if $Cat_{c\otimes}$ is locally $2$-presentable there? Does this sound reasonable? –  Martin Brandenburg Jun 3 '11 at 22:35
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@2) There's a proof in my second book (Lemma 6.3.7.9). @3) In a larger universe, the collection of locally U-presentable categories will be small, not locally presentable. (You can take U-small colimits, but not arbitrary colimits.) –  Jacob Lurie Jun 4 '11 at 16:22
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@Theo, my suggestion would be that you adjust your expectations. (-: There's a general notion of "locally presentable V-category" for any enriching category V, and I would expect locally presentable 2-categories to be the Cat-enriched version of that (possibly weakened). I think "locally locally presentable 2-category," whose hom-categories are locally presentable, would be a different sort of beast. –  Mike Shulman Jun 6 '11 at 6:30
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BTW, the problem with dropping "locally" is that there's a different extant notion of "presentable category," namely one presented by generators and relations, and this is the one which matches the general notion of "presentation" in universal algebra. –  Mike Shulman Jun 6 '11 at 6:34

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