What is a most natural categorification of a vector space?

Question

Few days ago I became excited when I learned from an answer to Examples of simple vertex operator algebras (VOAs) that

The irreducible modules of the rank $d$ free boson are naturally parametrized by points in $d$-dimensional space.

First thing that I would like to see, actually, is the details of the above. But then, VOAs are scary. I wonder if there is more concise and natural way to do it.

So, the main question is:

For a vector space $V$ over any field $k$, what is the most natural triangulated category $\mathscr T$ with $K_0(\mathscr T)$ isomorphic to $V$?

I am well aware that unless $k$ is a prime field or "something like" $\mathbb R$, one has to additionally specify some structure on $\mathscr T$ which will incorporate scalar multiplication by $k$ on $K_0(\mathscr T)$ but, well, this is part of the question.

It would be wonderful to have for $\mathscr T$ the derived category of one or other kind (like of perfect complexes over an algebra, or some bounded/unbounded derived category thereof, or the same with coherent sheaves over some scheme, etc. etc.) but I do not insist on that, any natural and "well-composed" example would be interesting, like some particular category of spectra or something like that.

There most probably are "trivial" examples although even for them I don't see it completely: I thought about modules over the product of $k$ copies of $k$; this would most probably work for finite $k$ but for infinite $k$ I think one obtains something different; while the sum of $k$ copies of $k$ does not have a unit. Maybe adjoining a unit would do it, I don't know. In short, the question is not protected from such trivial killing, and having one of the answers along these lines cannot be possibly avoided. But... well, I will wait for other answers.

I must also mention that there are several interesting examples when one can reach the "nonnegative part" of this question. Like, an answer to Categorifying the Reals via von Neumann Algebras? refers to an answer to another question where groupoid cardinalities are proposed as categorifications of positive rationals.

The oldest instance of this phenomenon that I know I learned from "Algebraic K-theory" of Bass: in § 7 of Chapter IX he investigates the category he calls $\mathbf{FP}(A)$. This is the symmetric monoidal category of faithfully projective modules over a commutative ring $A$, with the monoidal structure given by $\otimes_A$. Bass calculates $$ K_0(\mathbf{FP}(A))\cong U^+(\mathbb Q\otimes K_0(A)), $$ where $U^+$ means restricting to elements of strictly positive rank. Moreover, $$ K_1(\mathbf{FP}(A))\cong\mathbb Q\otimes K_1(A). $$ I wonder if this can be used somehow to produce good $K_0$'s that are $\mathbb Q$-vector spaces. I have vague feeling that maybe lambda-rings are relevant here but do not see anything definite about it.

Another possible source might be the fact that higher $K$-groups of algebraically closed fields are divisible. Could some forms of deloopings or versions of Bott periodicity be used to shift this to $K_0$ somehow?

But also it would be very interesting to see naturally occurring $K_0$'s that are vector spaces over finite fields.

And also $K_0$'s with natural action of $\sqrt{-1}$ too.

For $\mathbb R$, there are several interesting possibilities contained in answers to Categorifications of the real numbers, I wonder if any of those can be naturally extended to other fields.

Answer 1

I don't have an answer to your other questions, but I can address the first question. The vertex algebra for the rank $d$ free boson is (after forgetting some structure) a graded vector space whose weight 1 subspace is naturally a $d$-dimensional abelian Lie algebra. In this setting (in particular, because of the "freeness"), modules for the vertex algebra are in natural bijection with representations of this Lie algebra. Irreducible representations are nicely parametrized by points in $d$-dimensional space, but the extensions can be a bit of a nightmare. There is a monoidal structure on modules, which amounts to addition for the parameters attached to irreducible modules. There is also a braiding coming from a quadratic form that we fix in the beginning, but let us ignore that for now.

In conclusion, this particular category of modules is more tractably written as one of the following:

1. finite dimensional representations of an abelian Lie algebra
2. finite dimensional representations of a polynomial ring
3. coherent sheaves of finite length on affine space

Answer 2

One part of your question asks for naturally occuring categories (which are triangulated, presumably, since triangulated categories are the categorical setting for earlier parts of the question) whose $K_0$-groups are vector spaces over finite fields.

One source of such triangulated categories is stable module categories of quasi-Frobenius rings. Recall that the "stable module category" of a ring $R$ has, as objects, the $R$-modules, and as morphisms from $M$ to $N$, the set of equivalence classes of $R$-module morphisms $M\rightarrow N$, where two morphisms are said to be equivalent if their difference factors through a projective $R$-module. Recall also that a ring is "quasi-Frobenius" if its projective modules coincide with its injective modules. The stable module category of a quasi-Frobenius ring $R$ has a natural triangulation; I will write $stMod(R)$ for that triangulated category, restricted to its finitely generated modules, in order to avoid an Eilenberg swindle argument which would cause $K_0$ to vanish.

Consider the example $R = \mathbb{Z}/4\mathbb{Z}$. Every finitely-generated $R$-module is a direct sum of copies of $\mathbb{Z}/2\mathbb{Z}$ and $P = \mathbb{Z}/4\mathbb{Z}$, and $P$ is zero in the stable module category of $R$. Hence $K_0(stMod(R))$ must be a cyclic group generated by $[\mathbb{Z}/2\mathbb{Z}]$. However, we have the short exact sequence $$ 0 \rightarrow \mathbb{Z}/2\mathbb{Z}\rightarrow P \rightarrow \mathbb{Z}/2\mathbb{Z} \rightarrow 0$$ of $R$-modules, and consequently $[\mathbb{Z}/2\mathbb{Z}] + [\mathbb{Z}/2\mathbb{Z}] = [P] = 0$ in $K_0(stMod(R))$. So we have $K_0(stMod(R)) \cong \mathbb{Z}/2\mathbb{Z}$.

By a similar argument, for any prime $p$ and any positive integer $n$, we have $K_0(stMod(\mathbb{Z}/p^n\mathbb{Z})) \cong \mathbb{Z}/n\mathbb{Z}$. We also have $K_0(stMod(k[x]/x^n)) \cong \mathbb{Z}/n\mathbb{Z}$ for any field $k$. All that is really necessary for these kinds of $K_0$ calculations is that the ring $R$ be quasi-Frobenius, so that its stable module category is triangulated, and that $R$ is an Artinian principal ideal ring, so that its modules are all direct sums of cyclics.

Consequently, if $p$ is a prime and you want to recover $\mathbb{F}_p^n$ as $K_0$ of a triangulated category, then the triangulated category $stMod(R)$ will work, where $R$ is a Cartesian product of $n$ copies of $\mathbb{Z}/\ell^p\mathbb{Z}$ for any prime $\ell$ (or, alternatively, $n$ copies of $k[x]/x^p$ for any field $k$).

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You also ask about constructing triangulated categories whose $K_0$ is a vector space over, for example, the rational numbers. This is much more difficult than the finite field case. It is a nontrivial task to construct a triangulated category whose $K_0$ is $\mathbb{Q}$, or even just $\mathbb{Z}[\frac{1}{2}]$; this is tied up in the general problem of figuring out how to modify a given triangulated category so that a prime becomes inverted in its $K_0$. The papers https://arxiv.org/abs/1610.07162 and https://arxiv.org/abs/1702.07466 are about this problem, and you can find positive results specifically about the $\mathbb{Q}$ case of your problem in the Barwick-Glasman-Hoyois-Nardin-Shah paper (the first arXiv link).