Symmetric Monoidal Bicategories
Posted by Tom Leinster
Guest post by Nick Gurski
Angélica Osorno and I recently posted a preprint of our paper
on the arXiv. This is a project that she and I have been collaborating on for almost a year and a half, and I am very excited that we finally were able to prove the big coherence result:
Theorem: In a symmetric monoidal bicategory, every* diagram of constraint 2-cells commutes.
Below the fold, I will tell you a little bit about what a symmetric monoidal bicategory is, what this coherence theorem means if you are working with one, and why we were interested in proving this theorem in the first place.
A symmetric monoidal bicategory is supposed to be exactly what the name suggests: a bicategory equipped with a tensor product that is commutative in some appropriate sense. Let’s break that down into the relevant layers of structure without getting into too much detail.
- We start with a bicategory $B$.
- It has a monoidal structure, so we get a tensor product $x \otimes y$ of objects, of 1-cells, and of 2-cells. This comes with an associator and left and right unit constraints, but these are now equivalences instead of the usual isomorphisms we are used to from monoidal categories. We also need some invertible 2-cells to take the place of the usual associativity pentagon ($\pi$) and unit triangle ($\mu$), as well as two others that correspond to the “extra” unit axioms from the original definition of a monoidal category. (These last 2-cells are usually called $\lambda, \rho$.)
- $B$ has a braided structure which gives an equivalence $R_{x y}:x \otimes y \rightarrow y \otimes x$. There are also invertible 2-cells between the two obvious composites $x y z \rightarrow y z x$ and similarly for $x y z \rightarrow z x y$. (These 2-cells are called $R_{x|y z}$ and $R_{x y|z}$.)
- There is a syllepsis which is an isomorphism $v_{x y}:R_{y x}R_{x y} \cong 1_{x y}$.
- Finally being symmetric imposes just one additional axiom which says that the two different isomorphisms $R_{x y}R_{y x}R_{x y} \cong R_{x y}$ (one uses $v_{x y}$, the other $v_{y x}$) are equal.
This is a fairly complicated piece of algebra, so we wanted a coherence theorem that would do most of the calculational work for us.
Just as many naturally-occurring categories have symmetric monoidal structures, the same is true at the bicategorical level. The 2-category Cat has a symmetric monoidal structure given by the cartesian product, and the related bicategory Prof of categories, profunctors, and transformations has a symmetric monoidal structure. You can do all of this in the enriched world to get things like a symmetric monoidal structure on the bicategory of rings, bimodules, and bimodule homomorphisms. Mike Shulman has a nice preprint up that explains how you can construct many of the symmetric monoidal structures on these bicategories.
Our coherence theorem states the following. Start with a symmetric monoidal bicategory, and paste together constraint 2-cells (those are the 2-cells $\pi, \mu, \lambda, \rho, R_{-|--}, R_{--|-}, v$ plus things like naturality 2-cells) in any way you like. Now do this again, making sure your second pasting has the same source and target as the first. Then the two different 2-cells you constructed are in fact equal. This result is, in some ways, completely different from the coherence theorem for symmetric monoidal categories, but nevertheless it is what we expect to happen. The coherence theorem for symmetric monoidal categories states that if you build a pair of parallel morphisms out of the coherence constraints in that structure, then they are equal if they have the same underlying permutation. In our case, the permuting happens at the level of 1-cells: the source and target 1-cells of every coherence 2-cell already have the same permutation. So in a sense you should think of a 1-cell built out of the coherence 1-cell constraints as a kind of presentation for a particular permutation, and the 2-cell isomorphisms between them serve to tell you that any two presentations for the same underlying permutation are on equal footing in the sense that they are uniquely isomorphic. (The * in the statement of the theorem is the usual caveat that really we have to work with free symmetric monoidal bicategories, or at least diagrams which come from free ones.)
What are some applications of this coherence theorem (aside from making computations in one much simpler)?
- Eugenia Cheng and I got interested in thinking about categories weakly enriched in a monoidal bicategory, and more specifically the totality of such things. You can make this into a bicategory by using an enriched version of icons; I talked about this all the way back in CT2009 in Genova. Well, coherence for symmetric monoidal bicategories is exactly the theorem you need to show that if you enrich in a symmetric monoidal bicategory, the total structure you get back out is also a symmetric monoidal bicategory.
- You can also use this to show that the classifying space of a symmetric monoidal bicategory has an $E_{\infty}$ structure. We do this by constructing a pseudo-$\Gamma$-bicategory (this is a weak functor of tricategories from Segal’s category $\Gamma^{op}$ to the tricategoy of bicategories), then use a kind of Grothendieck construction to produce an actual $\Gamma$-bicategory (this is a functor of categories $\Gamma^{op} \rightarrow \mathbf{Bicat}$), and then take the classifying space to get a $\Gamma$-space.
There are a wide variety of things one could try to prove next.
- One of the main reasons we got interested in this result was how it should help if you want to try to prove that Picard 2-categories model stable homotopy 2-types. A Picard 2-category is a symmetric monoidal bicategory in which every 2-cell is invertible, every 1-cell is an equivalence, and every object has a tensor inverse (up to equivalence). It is a kind of twice-categorified version of an abelian group. Our coherence theorem, as well as the construction of the $E_{\infty}$ structure on the classifying space, are the first steps towards proving that there is some kind of equivalence (probably the easiest thing to do is consider the homotopy categories, or maybe try to give a Quillen equivalence of model structures) between Picard 2-categories and stable homotopy 2-types.
- Another interesting next step is to see how operads play into this picture. We used $\Gamma$-spaces to get an $E_{\infty}$ structure instead of operad actions, but you should be able to use either machine. We started thinking about this question, and even proved some preliminary results, but in order to get the constructions that we really wanted it seemed necessary to think about pseudo-algebras over operads or pseudo-algebras over pseudo-operads. That gets into some complicated, but very intriguing, territory using 2-monads, pseudomonads, and a whole host of 2-dimensional algebra.
- On the purely categorical side, there is still the open question of coherence for sylleptic monoidal bicategories. In every other case (monoidal, braided, or symmetric), the coherence theorem states that any diagram of 2-cells will commute, but in the sylleptic case that will no longer be true. A good coherence theorem in this case would give some criterion on how a diagram of 2-cells is constructed in order to ensure that it commutes. I have a feeling that one could prove such a theorem using the kind of strategy I used for the braided case, but that requires doing some nontrivial geometry first.
Re: Symmetric Monoidal Bicategories
This is great!! Exciting to see so much progress being made on this sort of thing.
One thing I briefly wondered about reading the blog post, but which you answered in the paper: there is also an aspect to the coherence theorem that talks about 1-cells, namely two parallel composites of constraint 1-cells are isomorphic iff they have the same underlying permutation (and in that case the isomorphism is unique, by the 2-cell part of the theorem).
Richard Garner and I are also working on a paper that will involve (bi)categories (weakly) enriched in a monoidal bicategory, and specifically weighted (bi)limits in such things. Our current version doesn’t assemble them into a categorical totality, but I would like to know that they aren’t just a bicategory (with icons) but also a tricategory. Have you thought about moving up to that level?
Finally, the question of what the right coherence theorem for the sylleptic case might be is intriguing!