### Symmetric Monoidal Bicategories and (n×k)-categories

#### Posted by Mike Shulman

I just put a note on the arXiv about constructing symmetric monoidal bicategories. The definition of symmetric monoidal bicategory is somewhat imposing, but in many cases all the structure can be “lifted” for free from a symmetric monoidal double category, which is a much easier structure. For instance, the bicategories of spans, cobordisms, and profunctors all inherit their monoidal structures in this way. Here’s the link:

The tricky part is that in a monoidal double category, the coherence isomorphisms such as $X\otimes (Y\otimes Z)\cong (X\otimes Y)\otimes Z$ are *vertical* isomorphisms, but these are the morphisms that get discarded when we pass to the horizontal bicategory. Thus, we need to be able to “lift” these isomorphisms to horizontal equivalences, in such a way that we can ensure coherence is preserved. The structure we need is essentially that of a proarrow equipment (regarded as a double category), although in this context, with attention focused on the “bicategory of proarrows” rather than the 2-category of arrows, it seems more appropriate to call it a framed bicategory or a *fibrant double category*. Richard Garner and Nick Gurski proved essentially the same result in section 5 of this paper, using “locally-double bicategories;” the machinery I used also makes braiding and symmetry easy to deal with.

More generally, I suspect that there is a theorem along the following lines. Define an $(n\times k)$-category (read “$n$ by $k$ category”) to be an $n$-category internal to $k$-categories. Such a thing has $(n+1)(k+1)$ different kinds of cells which are naturally arranged in a grid, with $(0,0)$-cells at the bottom left (say), $(n,0)$-cells at the top left, and $(n,k)$-cells at the top right. In particular, a double category is a 1x1-category. The theorem I think should be true is that if you have an $(n\times k)$-category which is “fibrant” in a suitable sense, then by discarding most of those types of cells and considering only the $(n+k+1)$ ones that go up the left edge and across the top edge of the rectangle, you should obtain an ordinary $(n+k)$-category.

The notion of $(n\times k)$-category may seem abstruse, but they actually occur quite naturally. Let’s unpack the definition a bit more. It’s easiest to imagine in the case $k=1$, where an $(n\times 1)$-category is just an internal $n$-category in 1-categories, so it comes with $(n+1)$ different categories $D_0, \dots, D_n$ related by source, target, unit, and composition maps (this is what Batanin calls a *monoidal $n$-globular category*). Now we are supposed to get an $(n+1)$-category by discarding the morphisms of $D_0, D_1,\dots,D_{n-1}$ and keeping only their objects, but also keeping both the objects and morphisms of $D_n$.

When $n=1$, this just means discarding the vertical arrows in a double category to get a bicategory. When $n=2$, a 2x1-category becomes a little more complicated, but if $D_0$ is the terminal category, then it reduces to simply a monoidal double category. Thus, the actual theorem I mentioned above is a special case of the putative theorem for 2x1-categories. But there are also non-degenerate 2x1-categories, such as the one in which $D_0$ consists of commutative rings and ring homomorphisms, $D_1$ consists of two-sided algebras and algebra homomorphisms, and $D_2$ consists of bimodules and bimodule maps. The underlying tricategory of this consists of commutative rings, two-sided algebras, bimodules, and bimodule maps. I believe that the “conformal nets” of Arthur Bartels, Chris Douglas, and André Henriques are supposed to be a similar structure but which is only “braided” at the bottom level, rather than fully commutative.

We can also write down some naturally occurring 1x2-categories; such a thing consists of 2-categories $D_0$ and $D_1$ related by source, target, unit, and composition maps. Suppose we take $D_0$ to be the 2-category of monoidal categories, monoidal functors, and transformations, and $D_1$ to consist of “bimodules” over these—not in the sense of profunctors, but meaning categories with an associative action on both sides by a pair of monoidal categories. Then we have a 1x2-category, which again has an underlying tricategory consisting of monoidal categories, bimodules, bimodule functors, and bimodule transformations.

Finally, this sort of thing also occurs in $(\infty,n)$-category theory. In particular, a Segal space can be viewed as a (weak) internal category in spaces, i.e. a $(1\times (\infty,0))$-category. The “completion” process which takes a Segal space to a complete Segal space, which is a sort of $(\infty,1)$-category, can then be viewed as a version of the above theorem (bearing in mind that $1 + (\infty,0) = (\infty,1)$). Similarly, an $n$-fold Segal space is like an internal $n$-fold category in spaces, i.e. a $(1\times 1\times \dots\times 1\times (\infty,0))$-category, and “completion” associates to this its corresponding $1+1+\dots+1+(\infty,0) = (\infty,n)$ category.

In all of these cases, you can probably already guess that I think it’s often better to just work with the $(n\times k)$-category as the fundamental structure. However, sometimes you really do want to forget the extra structure and just think about the underlying $(n+k)$-category. For instance, the Baez-Dolan cobordism hypothesis asserts some universal property of the $n$-category (or $(\infty,n)$-category) of cobordisms. If we want to construct a particular invariant of manifolds, i.e. a map out of this $n$-category, then we need to fix an $n$-category as the target. Now one way to construct the $(\infty,n)$-category of cobordisms is via completion of an $n$-fold Segal space, and the target $n$-category will also often come from an $((n-1)\times k)$-category for some $k$. But the universal property belongs only to the $(\infty,n)$-category of cobordisms itself, so the invariant we want only lives (a priori) at the level of $(\infty,n)$-categories. Thus, it’s important to understand the construction which takes an $(n\times k)$-category to an $(n+k)$-category.

## Re: Symmetric Monoidal Bicategories and (n x k)-categories

Trivial comment:

(n,0)-cells at the top left

I would have expected the top left to have `x-coordinate’ 0

or is this another example of categorical contrariness?