Kapranov and Voevodsky introduced the following 2-category $2{\rm Vect}$ of "2-vector spaces" over a field $K$:

- 0-cells are natural numbers $\langle N\rangle,\langle M\rangle...$
- 1-cells $\langle N\rangle\to \langle M\rangle $ are strictly-positive-integer valued matrices of size $M\times N$
- 2-cells $A\to B$ are matrices whose entries are matrices, whose $(i,j)$ entry is a matrix having size $A_{ij}\times B_{ij}$.

So for example, this is a 2-cell $\langle 2\rangle\to \langle 2\rangle$ where all the entries in $\alpha$ belong to the field $K$.

Since this is the best of possible worlds, this categorifies correctly (at least part of) the idea of a vector space as a "set of tuples made of scalars $a_i\in K$".

In fact, this is a way to make the following intuitive idea work:

just as the category of vector spaces is equivalent to the category of natural numbers, where $\hom(n,m)$ is the space of $m\times n$ $K$-valued matrices, so the 2-category of 2-vector spaces is equivalent to the 2-category of natural numbers, where $\hom(n,m)$ is the category of $K$-linear functors ${\rm Vect}^m \to {\rm Vect}^n$.

Now.

The 2-category of 2-vector spaces has two monoidal structures $\boxtimes$ and $\boxplus$ where

- $\langle N\rangle\boxtimes \langle M\rangle = \langle N\cdot M\rangle$;
- the integer-valued matrix $A\boxtimes B$ has in its $(ik,jl)$ entry the integer $A_{ik}B_{jl}$;
- given 2-cells $\alpha,\beta$ the 2-cell $\alpha\boxtimes\beta$ is the entry-wise tensor product of matrices.

and $\boxplus$ (the monoidal sum) is defined similarly, so that $\langle N\rangle ⊞ \langle M\rangle = \langle N+M\rangle$, $A ⊞ B$ is the direct sum of matrices, and $\alpha ⊞ \beta$ is the entry-wise direct sum of matrices (in fact $2{\rm Vect}$ a rig category with respect to $\boxtimes$ and $\boxplus$).

Question: Is the $\boxtimes$-structure also

closed?

Representing the category $2{\rm Vect}(\langle mn\rangle,\langle p\rangle)$ as the category of linear functors $\underbrace{{\rm Vect}^n \times\dots\times{\rm Vect}^n}_{m \text{ times}} \to {\rm Vect}^p$ does not seem to help, and before going deep down the computations I would like to know if there is a conceptual reason for this category to (not) be closed.