# Is there a symmetric monoidal 2-category "SuperDuperVect"?

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.

• My guess would be that the 'symmetry structure' in SuperDuperVect comes from an appropriate truncation of the sphere spectrum, since that is where it comes from for SuperVect. Jun 20, 2015 at 1:03
• @DavidRoberts I don't think I know that story. Will you elaborate? Jun 21, 2015 at 0:21
• As to where to look up the axioms, see ncatlab.org/nlab/show/monoidal+bicategory#references Jun 21, 2015 at 15:30
• @MikeShulman Thanks! I should have remembered that Chris's thesis has a definition. Also, I think you have some work along these lines for the Morita bicategory? Jun 21, 2015 at 19:00
• @Theo The free Picard category on one object is the category of Z-graded Z-lines. This is the truncation of the sphere spectrum to dimensions 0 and 1. If one instead takes the looping of the sphere spectrum and then truncates to dimensions 0 and 1 (so dimensions 1 and 2 in the unlooped case) then one has a similar structure, but now with Z/2-graded (=super) lines. The categorified sign representation takes values 'in this category' (see Ganter-Kapranov arXiv/1110.4753). The symmetry structure I think extends to that on supervector spaces more generally. Work of Osorno-Gurski is also relevant. Jun 21, 2015 at 23:19

Yes, the symmetric monoidal 2-category you are looking for does exist.

I think that there is a slightly different 2-category which is better, but yours embedds inside the one I will describe, which differs in that there are interesting "cross-terms" from $\Xi$ to 1, i.e. morphisms between these objects. This 2-category has a more familiar description. It is the Morita category of finite dimensional semisimple superalgebras (over $\mathbb{C}$).

Here a superalgebra is just an algebra object in SuperVect. The theory of semisimple modules and algebras mirrors that for ordinary algebras, but with a few subtleties (it is super subtle). A very nice treatment which covers the case that the ground field is algebraically closed is this paper:

Semisimple Superalgebras Tadeusz Jozefiak Volume 1352 of the series Lecture Notes in Mathematics pp 96-113.

There you see that there is a classification of semisimple superalgebras over $\mathbb{C}$. They are finite sums of simple superalgebras. The simple superalgebras are classified as either

1. $End(\mathbb{C}^{p|q})$ (which is super Morita equivalent to $\mathbb{C}$)
2. $Q(n) = M_n(\mathbb{C}) \otimes Cl_1$ where $Cl_1$ is the Clifford algebra on a one-dimensional complex vector space.

They have multiplications (using the super-tensor product, of course):

1. $End(\mathbb{C}^{p|q}) \otimes End(\mathbb{C}^{m|n}) = End(\mathbb{C}^{pm + qn|pn + qm})$
2. $End(\mathbb{C}^{p|q}) \otimes Q(n) = Q(pn + qn)$
3. $Q(m) \otimes Q(n) = End(\mathbb{C}^{mn|mn})$

Now the 2-category I want to consider is the Morita 2category whose objects are fin. dim. semisimple superalgebras and whose 1-morphisms are superbimodules between them.

The isomorphism classes of objects in this 2-category are given by pairs of natural numbers which count the number of $End(\mathbb{C}^{p|q})$ and $Q(n)$ factors.

The categories of morphisms are exactly what you describe, as long as you throw out the cross-term morphisms. For example the category of morphisms from $\mathbb{C}$ to $Q(n)$ is equivalent to the category of $Q(n)$-modules, where in your 2-category (if I understand your notation) you would have the zero category here. The 2-category you describe sits inside as the sub-2-category with only the zero cross term morphisms.

Now for your final question. There is no symmetric monoidal functor $$SuperDuperVect \to Vect_{SuperVect}$$ where SuperDuperVect is either the Morita category I describe or the subcategory you mention. You can see this by passing to maximal Picard sub-2-categories (i.e. the max 2-category in which all morphisms and objects are invertible).

These Picard 2-categories are equivalent to spectra with 3 consecutive homotopy groups. The target gives a spectrum with $$\pi_0 = 0, \; \pi_1 = \mathbb{Z}/2, \; \pi_2 = \mathbb{C}^\times$$ while the source (in either case) has $$\pi_0 = \mathbb{Z}/2, \; \pi_1 = \mathbb{Z}/2, \; \pi_2 = \mathbb{C}^\times$$. Moreover the k-invariants of these are well-known. The latter looks like a truncated variant of the Brown-Comenetz dual of the sphere. It is closely related to real K-theory KO. The former (the target) looks like the connective cover. The k-invariant connecting the bottom $\mathbb{Z}/2$ to the other homotopy groups (eg. the next $\mathbb{Z}/2$) obstructs the existence of your map, and it is known to be non-zero.

• See also the references at ncatlab.org/nlab/show/super+line+2-bundle Nov 17, 2015 at 16:35
• This answer reminded me of a lot of things that I knew about but had forgotten. But now I'm not sure I believe it. Actually, the problem is probably my fault: either I am simply confused, or I chose bad notation. My notation "Vect_SuperVect" is not intended to mean "categories that are direct sums of copies of SuperVect", but rather "module categories for SuperVect" (so perhaps I should have used "Mod_SuperVect"), with the balanced tensor product. Now, surely the Morita category of superalgebras embeds in here just as in the usual case? E.g. Cliff(1) corresponds to Vect as a SuperVect-module. Nov 18, 2015 at 15:27
• But I will say: it had never occurred to me to ask what the symmetry of Cliff(1) with itself is. That it is the odd Cliff(2)-bimodule is interesting! Nov 18, 2015 at 15:30
• This is a super answer. Nov 28, 2015 at 20:49