October 4, 2012

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.
Posted at October 4, 2012 4:15 PM UTC

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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!

Posted by: Mike Shulman on October 7, 2012 2:57 AM | Permalink | Reply to this

Re: Symmetric Monoidal Bicategories

Thanks Mike! The 1-cell part of the theorem is a good example of something that we struggled with throughout the proof, but that might be largely invisible now. That is the question of how to translate the categorical information we wanted into combinatorial information in such a way that the problems would be easy. Our proofs rely quite heavily on exploiting the braided structure, so we were always looking for ways to turn 1- and 2-cells in symmetric monoidal bicategories into braids so that the resulting geometry/group theory was something people already knew about. We settled on replacing everything with its positive braid part, and that had a much wider impact than I was expecting at first. In particular, the proof that parallel 1-cells with the same underlying permutation are isomorphic falls out of that fairly easily.

As for a larger structure of symmetric monoidal bicategories, I scribbled down the whole thing a long time ago. Eugenia and I published the case for plain monoidal bicategories, and the symmetric case is basically the same although a bit more work. I am quite interested to see what you and Richard come up with, though, sounds like a very nice paper.

One more thing I did think of related to this whole picture is the question of putting a Thomason-style model structure on symmetric monoidal bicategories so that they model connective spectra. That seems like a nice project to give a PhD student.

Posted by: Nick Gurski on October 8, 2012 8:58 AM | Permalink | Reply to this

Re: Symmetric Monoidal Bicategories

Hmm, I thought in the post you were talking about the totality of $V$-enriched (bi)categories for a fixed monoidal bicategory $V$, not the totality of monoidal bicategories. Maybe I misunderstood.

Could you prove the part of the coherence theorem about 1-cells by just looking at the proof for ordinary braided or symmetric monoidal 1-categories and replacing every use of an axiom with a 2-cell?

Posted by: Mike Shulman on October 10, 2012 2:15 AM | Permalink | Reply to this

Re: Symmetric Monoidal Bicategories

Does this work reassure us that whatever the right form of string diagrams is (surface diagrams in at least 6d?), they will work for symmetric monoidal bicategories?

Posted by: David Corfield on October 8, 2012 8:58 AM | Permalink | Reply to this

Re: Symmetric Monoidal Bicategories

I think that is correct, yes. My proof in the braided case essentially is some form of surface diagram argument, but here we worked purely algebraically. But the same ideas seem to work just fine: if you look at surfaces in $\mathbb{R}^{6}$, then those should be the 2-cells for something which is symmetric monoidal biequivalent to the free gadget, which says that you can compute in your favorite symmetric monoidal bicategory using surfaces. I think in the sylleptic case, a surface diagram interpretation is really the way to go, but then you want some concrete way to think about that algebraically which is where I get completely lost.

Posted by: Nick Gurski on October 8, 2012 9:12 AM | Permalink | Reply to this

Re: Symmetric Monoidal Bicategories

This is really exciting! I have been asking people whether something like this is true for a long time. I’d heard on the grapevine that you were working on a proof, so it’s good to see it at last.

One major reason I’m interested in this is because of work with finitely-presented symmetric monoidal bicategories, produced from generating 0-cells, generating 1-cells, generating 2-cells, and relations between composites of generating 2-cells. It is good to do this in such a way that the result is completely weak. But the relations between composites of generating 2-cells in principle need to include all relevant constraint 2-cells — and then without a theorem like this, there is no way to be sure that these constraint 2-cells have been introduced in the ‘correct’ way.

I’m also interested in the algorithmic side of this. Suppose you have two isomorphic 1-cells in a free symmetric monoidal bicategory. Is there some terminating procedure that will construct for you a 2-cell going between them? In particular, is there a normal form for such 1-cells?

Posted by: Jamie Vicary on October 8, 2012 11:21 AM | Permalink | Reply to this

Re: Symmetric Monoidal Bicategories

Thanks for the interest, Jamie. The question of the algorithmic side of things is quite interesting. Here is what we prove in the paper.

Start with a pair of 1-cells with the same underlying permutation. Now we only work with 1-cells that have an underlying braid which is a positive braid; if you have a 1-cell which does not have a positive braid, then it is fairly easy to fix it so that it does. We then use a particular instance of what is called a left-weighted factorization of a positive braid to write this in a special form, and then we are able to remove an $R^{2}$ with the syllepsis. This factorization is a purely combinatorial gadget, and there is an algorithm that takes a braid and gives you the left-weighted factorization. Keep doing this, and you eventually get down to what we call a minimal 1-cell, which just means that the underlying braid has the property that no two strands cross twice. That is the normal form for 1-cells.

On the level of 2-cells, there is also something like a normal form. Our coherence theorem says that every 2-cell $f \Rightarrow g$ is actually a composite $f \Rightarrow m \Rightarrow g$ where $m$ is a minimal 1-cell and the 2-cells in this factorization are the ones I constructed above. In other words, every 2-cell can be written as a composite of one which reduces double crossings at the braid level all the way down as far as possible and then one that builds double crossings back in again. Now how you might get a computer to actually do any of this for you is beyond me, but I have long been of the opinion that someone should be able to write a program to do pasting diagram computations and then we could suddenly prove all sorts of theorems extremely easily.

Posted by: Nick Gurski on October 9, 2012 12:42 PM | Permalink | Reply to this

Re: Symmetric Monoidal Bicategories

Dude, I think Genova was CT2010, and that’s where you talked about the Iterated Icon thing. I know we’ve been taking a while writing it up, but not *that* long. And we’re getting there!

Posted by: Eugenia Cheng on December 12, 2012 6:01 PM | Permalink | Reply to this

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