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September 10, 2014

Quasistrict Symmetric Monoidal 2-Categories via Wire Diagrams

Posted by Simon Willerton

Guest post by Bruce Bartlett

I recently put an article on the arXiv:

It’s about Chris Schommer-Pries’s recent strictification result from his updated thesis, that every symmetric monoidal bicategory is equivalent to a quasistrict one. Since symmetric monoidal bicategories can be viewed as the syntax for ‘stable 3-dimensional algebra’, one aim of the paper is to write out this stuff out in a diagrammatic notation, like this:

pic

The other aim is to try to strip down the definition of a ‘quasistrict symmetric monoidal bicategory’, emphasizing the central role played by the interchangor isomorphisms. Let me explain a bit more.

Motivation

Firstly, some motivation. For a long time now I’ve been finishing up a project together with Chris Douglas, Chris Schommer-Pries and Jamie Vicary about 1-2-3 topological quantum field theories. The starting point is a generators-and-relations presentation of the oriented 3-dimensional bordism bicategory (objects are closed 1-manifolds, morphisms are two-dimensional bordisms, and 2-morphisms are diffeomorphism classes of three-dimensional bordisms between those). So, you present a symmetric monoidal bicategory from a bunch of generating objects, 1-morphisms, and 2-morphisms, and a bunch of relations between the 2-morphisms. These relations are written diagrammatically. For instance, the ‘pentagon relation’ looks like this:

pic

To make rigorous sense of these diagrams, we needed a theory of presenting symmetric monoidal bicategories via generators-and-relations in the above sense. So, Chris Schommer-Pries worked such a theory out, using computads, and proved the above strictification result. This implies that we could use the simple pictures above to perform calculations.

Strictifying symmetric monoidal bicategories

The full algebraic definition of a symmetric monoidal bicategory is quite intimidating, amounting to a large amount of data satisfying a host of diagrams. A self-contained definition can be found in this paper of Mike Stay. So, it’s of interest to see how much of this data can be strictified, at the cost of passing to an equivalent symmetric monoidal bicategory.

Before Schommer-Pries’s result, the best strictification result was that of Gurski and Osorno.

Theorem (GO). Every symmetric monoidal bicategory is equivalent to a semistrict symmetric monoidal 2-category.

Very roughly, a semistrict symmetric monoidal 2-category consists of a strict 2-category equipped with a strict tensor product, plus the following coherence data (see eg. HDA1 for a fuller account) satisfying a bunch of equations:

  • tensor naturators, i.e. 2-isomorphisms Φ f,g:(fg)(fg)(ff)(gg)\Phi_{f,g} : (f' \otimes g') \circ (f \otimes g) \Rightarrow (f' \circ f) \otimes (g' \circ g)
  • braidings, i.e. 1-morphisms β A,B:ABBA\beta_{A,B} : A \otimes B \rightarrow B \otimes A
  • braiding naturators, i.e. 2-isomorphisms β f,g:β A,B(fg)(gf)β A,B\beta_{f,g} : \beta_{A,B} \circ (f \otimes g) \Rightarrow (g \otimes f) \circ \beta_{A,B}
  • braiding bilinearators, i.e. 2-isomorphisms R (A|B,C):(idR B,C)(R A,Bid)R A,BCR_{(A|B, C)} : (id \otimes R_{B,C}) \circ (R_{A,B} \otimes id) \Rightarrow R_{A, B\otimes C}
  • symmetrizors, i.e. 2-isomorphisms ν A,B:id ABR B,AR A,B\nu_{A,B} : id_{A \otimes B} \Rightarrow R_{B,A} \circ R_{A,B}

So — Gurski and Osorno’s result represents a lot of progress. It says that the other coherence data in a symmetric monoidal bicategory (associators for the underlying bicategory, associators for the underlying monoidal bicategory, pentagonator, unitors, adjunction data, …) can be eliminated, or more precisely, strictified.

Schommer-Pries’s result goes further.

Theorem (S-P). Every semistrict monoidal bicategory is equivalent to a quasistrict symmetric monoidal 2-category.

A quasistrict symmetric monoidal 2-category is a semistrict symmetric monoidal 2-category where the braiding bilinearators and symmetrizors are equal to the identity. So - only the tensor naturators, braiding 1-morphisms, and braiding naturators remain!

The method of proof is to show that every symmetric monoidal bicategory admits a certain kind of presentation by generators-and-relations (a ‘quasistrict 3-computad’). And the gismo built out of a quasistrict 3-computad is a quasistrict symmetric monoidal 2-category! Q.E.D.

Stringent symmetric monoidal 2-categories

In my article, I reformulate the definition of a quasistrict symmetric monoidal 2-category a bit, removing redundant data. Firstly, the tensor naturators Φ (f,g),(f,g)\Phi_{(f',g'),(f,g)} are fully determined by their underlying interchangors ϕ f,g\phi_{f,g},

(1)ϕ f,g=Φ (f,id),(id,g):(fid)(idg)(idg)(fid) \phi_{f,g} = \Phi_{(f, id), (id, g)} : (f \otimes id) \circ (id \otimes g) \Rightarrow (id \otimes g) \circ (f \otimes id)

This much is well-known. But also, the braiding naturators are fully determined by the interchangors. So, I define a stringent symmetric monoidal 2-category purely in terms of this coherence data: interchangors, and braiding 1-morphisms. I show that they’re equivalent to quasistrict symmetric monoidal bicategories.

Wire diagrams

The ‘stringent’ version of the definition is handy, because it admits a nice graphical calculus which I call ‘wire diagrams’. I needed a new name just to distinguish them from vanilla-flavoured string diagrams for 2-categories where the objects of the 2-category correspond to planar regions; now the objects of the 2-category correspond to lines. But it’s really just a rotated version of string diagrams in 3 dimensions. So, the basic setup is as follows:

pic

But to keep things nice and planar, we’ll draw this as follows:

pic

These diagrams are interpreted according to the prescription: tensor first, then compose! So, the interchangor isomorphisms look as follows:

pic

So, what I do is write out the definitions of quasistrict and stringent symmetric monoidal 2-categories in terms of wire diagrams, and use this graphical calculus to prove that they’re the same thing.

That’s good for us, because it turns out these ‘wire diagrams’ are precisely the diagrammatic notation we were using for the generators-and-relations presentation of the oriented 3-dimensional bordism bicategory. For instance, I hope you can see the interchangor ϕ\phi being used in the ‘pentagon relation’ I drew near the top of this post. So, that diagrammatic notation has been justified.

Posted at September 10, 2014 8:43 PM UTC

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11 Comments & 0 Trackbacks

Re: Quasistrict Symmetric Monoidal 2-Categories via Wire Diagrams

The pentagon relation looks like K_4, when the rooted binary trees describe interactions of RP^1’s. These end up being really helpful in studying twistor-string amplitudes.

Posted by: kneemo on September 12, 2014 12:23 AM | Permalink | Reply to this

Re: Quasistrict Symmetric Monoidal 2-Categories via Wire Diagrams

Could you give me a reference on this?

Posted by: Bruce Bartlett on September 18, 2014 11:25 PM | Permalink | Reply to this

Re: Quasistrict Symmetric Monoidal 2-Categories via Wire Diagrams

All this is great, Bruce. Congratulations to you and your team!

My one tiny complaint is terminological. What you’re calling a ‘symmetrizor’ is basically what Ross Street already called a ‘syllepsis’. (Actually I think his syllepsis is the inverse of your symmetrizor.) I think his term is better, since then there’s a column of the periodic table that goes:

bicategory

monoidal bicategory

braided monoidal bicategory

sylleptic monoidal bicategory

symmetric monoidal bicategory

The syllepsis is the big surprise in this column of the periodic table, just as the braiding is the big surprise in the previous column. I don’t think ‘symmetrizoric monoidal bicategories’ sounds as good as ‘sylleptic monoidal bicategories’.

But this is no big deal…

Posted by: John Baez on September 18, 2014 2:18 PM | Permalink | Reply to this

Re: Quasistrict Symmetric Monoidal 2-Categories via Wire Diagrams

Agreed, I should have said “syllepsis”. I don’t actually use the term “symmetrizor” in the article. It only appeared in the above post because I suddenly felt at the n-category cafe I should give names to the remaining coherence data in Gurski and Osorno’s strictification theorem. There’s more latitude for that here in an informal setting.

Posted by: Bruce Bartlett on September 18, 2014 11:32 PM | Permalink | Reply to this

Re: Quasistrict Symmetric Monoidal 2-Categories via Wire Diagrams

I said that the syllepsis is the ‘big surprise’ in the n=2n = 2 column of the periodic table of nn-categories. What I meant is that sylleptic monoidal bicategories provide the ‘new row’ in this column as compared to the n=1n = 1 column, where we jump straight from braided monoidal categories to symmetric monoidal categories.

Similarly, the braiding is the ‘big surprise’ in the n=1n = 1 column. The syllepsis is a kind of ‘meta-braiding’, in a way that can be made precise. I won’t bother here, except to say that for any nn, the ‘new row’ in the nnth column of the periodic table involves a meta-meta-meta-\cdots-braiding.

This makes it fun to compare the braiding and the syllepsis.

Chris’s theorem proves something I’d long suspected: every symmetric monoidal bicategory is equivalent to one where the syllepsis is trivial.

This seems a bit unlike the analogous situation in the previous column: not every symmetric monoidal category is equivalent to one in which the braiding is an identity morphism. But I’m not sure this is a fair comparison.

It would be helpful to get a nice supply of sylleptic but not symmetric monoidal bicategories, to understand these issues better. Thanks to the work of Sjoerd Crans, we can obtain these by taking the centers of braided monoidal bicategories. However, I don’t think they’re understood very well!

Sylleptic monoidal bicategories with duals should give invariants of 2d manifolds embedded in 5-space, just as braided monoidal categories with duals give invariants of 1d manifolds embedded in 3-space. This is the ‘almost stable’ situation: add one more dimension of space and the knotting goes away.

Posted by: John Baez on September 18, 2014 2:41 PM | Permalink | Reply to this

Re: Quasistrict Symmetric Monoidal 2-Categories via Wire Diagrams

Finally: I really like the term ‘stringent’ as a variant of ‘strict’ in this context. As you continue to develop new kinds of semistrict nn-categories, here are some other terms you can use:

stern, severe, harsh, uncompromising, authoritarian, firm, rigid, tough, austere, inflexible, unyielding, unbending, no-nonsense.

Posted by: John Baez on September 18, 2014 2:45 PM | Permalink | Reply to this

Re: Quasistrict Symmetric Monoidal 2-Categories via Wire Diagrams

I like ‘stringent’, too. But (I’m sorry) the other terms you suggest, most of them, I really don’t like at all. (Or perhaps you were being tongue-in-cheek?)

‘Firm’ would be good, ‘rigid’ not quite as good (because it’ already used for a different concept). ‘Harsh’ and ‘authoritarian’ and ‘uncompromising’ and ‘no-nonsense’ would be terrible in my opinion!

I wouldn’t mind something along the lines of ‘firm’ such as ‘tight’ or ‘trim’.

Posted by: Todd Trimble on September 18, 2014 10:11 PM | Permalink | Reply to this

Re: Quasistrict Symmetric Monoidal 2-Categories via Wire Diagrams

I’m sure John is making a joke here. But just to be clear - I was just using the term stringent to refer to a certain form of the definition of a quasistrict symmetric monoidal bicategory. I’m not defining a new mathematical structure, just trying to clearly delineate, for the purpose of the article, two different versions of the same structure. The real choice was Chris Schommer-Pries’s choice (fair enough, in my opinion) of the term quasistrict symmetric monoidal bicategory as something stricter than a semistrict symmetric monoidal bicategory.

Posted by: Bruce Bartlett on September 18, 2014 11:41 PM | Permalink | Reply to this

Re: Quasistrict Symmetric Monoidal 2-Categories via Wire Diagrams

Yes, I was making a joke. I’m just amused by the idea of papers showing that every tough, no-nonsense nn-category is equivalent to an authoritarian one, or that firm nn-categories can be made unyielding.

But if the variety of different strictness conditions on nn-categories continues to flourish — and I have no reason to think it won’t, or shouldn’t — it seems fine to me if some of the conditions not destined for greatness have goofy names. After all, they can’t all have great names; there just won’t be enough great names. So the choice would be between boring names and funny names.

This applies to names for mathematical concepts in general. The really important concepts get used a lot and deserve great names. The not-so-important concepts get used less, maybe just in a few papers, and deserve either boring or funny names. You can probably guess what I find more enjoyable.

In fact, as we all know, there’s even the danger of using up the great names before the great concepts have been found. Mathematicians are pretty good at eventually saying “oh, I didn’t mean to use such a great word for a concept that, in retrospect, wasn’t quite right”, but it’s a somewhat painful process, so there’s a case to be made for deliberately choosing names that aren’t great when you’re fumbling around in the initial stages of a subject.

For example, in the subject of kk-tuply monoidal nn-categories we have a lot of different kinds of ‘flab’, described by the different ‘ators’ Bruce listed—and infinitely many more he didn’t list. So, there’s a game that’s rather interesting to some of us, to see how many of these can safely be assumed to be trivial, in the sense that every kk-tuply monoidal nn-category is equivalent to one in which these are trivial. Eventually there will be a few clear ‘winners’, in which maximal amounts of flab of various kinds have been squeezed out… and eventually there may be a magnificent theory of this — unless people decide it’s better to learn to live with flab, because weight-reduction plans of this sort are inherently a waste of time, or ‘evil’. (I can see it either way.) But for the last few decades and maybe the next few we can expect to see some theorems that sound like:

Theorem. Every symmetric monoidal bicategory is equivalent to a semistrict one.

Theorem. Every symmetric monoidal bicategory is equivalent to a quasistrict one.

Theorem. Every symmetric monoidal bicategory is equivalent to a stringent one.

and similarly for other kk-tuply monoidal nn-categories.

And in the process of sorting things out, we’ll need some names for things that aren’t so great. So, it seems we get to choose either boring names or funny names.

Of course in academia there’s a great institutional pressure toward boring names, since boringness looks superficially similar to seriousness, and scholars need to convey the impression of doing serious stuff, since we’re getting paid a decent wage to do stuff that doesn’t instantly yield much profit. But mathematicians more than most other academics are willing to countenance funny names for things, probably because our seriousness (and even boringness) is less in question.

Posted by: John Baez on September 21, 2014 3:16 PM | Permalink | Reply to this

Re: Quasistrict Symmetric Monoidal 2-Categories via Wire Diagrams

Nice! Your “wire diagrams” remind me of the string diagrams that Kate Ponto and I used for indexed monoidal categories in this paper. In fact, our notation was adapted from a notation used for symmetric monoidal bicategories by Daniel Schaeppi, by way of the fact that indexed monoidal categories give rise to symmetric monoidal bicategories (and a version of our notation works for general symmetric monoidal bicategories as well). Our notation has two kinds of strings and nodes, one for describing 1-morphisms and one for describing 2-morphisms; while Daniel’s notation omits the strings for 1-morphisms and yours omits the strings for the 2-morphisms.

Posted by: Mike Shulman on September 23, 2014 12:51 AM | Permalink | Reply to this

Re: Quasistrict Symmetric Monoidal 2-Categories via Wire Diagrams

Yes you’re right it is definitely similar.

Posted by: Bruce Bartlett on September 29, 2014 11:12 AM | Permalink | Reply to this

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