Recall that the category $\mathrm{SuperVect}$, as a category, consists of pairs of vector spaces, thought of as formal direct sums $V \oplus W\,\Pi$, where $\Pi$ is the "odd line". (Called "$\Pi$" because tensoring with it is "parity reversal".) Indeed, there is a good notion of "direct sum of categories" in which case, as categories, we have $$ \mathrm{SuperVect} = \mathrm{Vect} \oplus \mathrm{Vect}\, \Pi$$ As a monoidal category, we declare that the one-dimensional vector space $\mathbf 1 \in \mathrm{Vect}$ is the monoidal unit and that $\Pi \otimes \Pi = \mathbf 1$, and give it the trivial associator (so that as a monoidal category $\mathrm{SuperVect}$ is the category of sheaves of vector spaces on $\mathbf Z/2$ with the convolution product; when $2$ is invertible, which I assume it is, this is also the category of representations of $\mathbf Z/2$).

The interesting part of $\mathrm{SuperVect}$ is its braiding/symmetry. A braiding $\sigma$ on this monoidal category is uniquely determined by its value on
$$ \mathbf 1 = \Pi \otimes \Pi \overset \sigma \longrightarrow \Pi \otimes \Pi = \mathbf 1 $$
which is just a number $\sigma_{\Pi,\Pi}$. The axioms of symmetric monoidal category force $\sigma_{\Pi,\Pi}$ to square to $1$, but do not force it to be $1$ itself. The symmetric monoidal category $\mathrm{SuperVect}$ is determined by declaring that $\sigma_{\Pi,\Pi}$ is the *other* number that squares to $1$, namely $-1$.

At least over $\mathbb C$, $\mathrm{SuperVect}$ has the following important property, due to Deligne. Not every symmetric monoidal category is tannakian over $\mathrm{Vect}$ --- in particular, there does not exist a symmetric monoidal functor $\mathrm{SuperVect} \to \mathrm{Vect}$ --- but every symmetric monoidal category (satisfying some technical bounds on growth rates of objects) is tannakian over $\mathrm{SuperVect}$.

My question is whether this trick can be repeated one dimension higher. Let $\mathrm{Vect}_{\mathrm{SuperVect}}$ denote the 2-category of all "supercategories", i.e. categories with an action by $\mathrm{SuperVect}$. (I'd rather not make this precise. Probably I should fill in some words like "abelian". I'd be perfectly happy just working with the 2-category whose objects are the natural numbers and whose morphisms are matrices filled in with supervector spaces, just like one can model $\mathrm{Vect}$ as the category whose objects are natural numbers and whose morphisms are matrices of numbers.) Then $\mathrm{Vect}_{\mathrm{SuperVect}}$ is symmetric monoidal (because $\mathrm{SuperVect}$ is) with unit object $\mathbf 1$, and with $\mathrm{End}(\mathbf 1) = \mathrm{SuperVect}$.

Then there's a perfectly good 2-category
$$ \mathrm{SuperDuperVect} = \mathrm{Vect}_{\mathrm{SuperVect}} \oplus \mathrm{Vect}_{\mathrm{SuperVect}} \, \Xi$$
where the letter $\Xi$ is just a formal symbol playing the role of $\Pi$ above. I would like to give this category a symmetric monoidal structure in which $\Xi \otimes \Xi = \mathbf 1$ but the braiding on $\Xi$ is the endomorphism
$$ \sigma_{\Xi,\Xi} = \Pi \in \mathrm{End}(\mathbf 1) = \mathrm{End}(\Xi \otimes \Xi). $$
The idea is that this is *the other* object that squares to $\mathbf 1 \in \mathrm{SuperVect}$.

Now, I should be a bit careful. Symmetric monoidal 2-categories, when written out in full detail, consist of quite a lot of data and coherence conditions. Perhaps there's some rule that says that $\sigma_{\Xi,\Xi}$ not only has to square to $\mathbf 1$ but also has to braid trivially with itself. I don't know, and I'm not sure where to look up the axioms.

Does such a symmetric monoidal 2-category "$\mathrm{SuperDuperVect}$" exist?

Is it symmetric-monoidally equivalent to some more basic thing, say the 2-category of sheaves of supercategories on something with convolution product, or the 2-category of supercategorical representations of something?

Is there a symmetric monoidal 2-functor $\mathrm{SuperDuperVect} \to \mathrm{Vect}_{\mathrm{SuperVect}}$?

Of course, if the answer to the first question is "no", then the other two are moot. If the answer to the first question is "yes", then I can't imagine positive answers to the other two, but maybe my imagination is faulty.

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