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Stream: learning: questions

Topic: Lax monoidal lax functor

Fabrizio Genovese (Oct 28 2020 at 12:57):

Hello all, I've the following question:
We know that there's a bicategory $\textbf{Span}$ where objects are sets and morphisms are spans. 2-cells are morphisms between spans. This bicategory also admits a monoidal structure that arises from the cartesian product in $\textbf{Set}$.
Now suppose I have another monoidal category $C$. I can see $C$ as a bicategory where all the 2-cells are trivial (note: This is different from seeing a monoidal category as a bicat with one object!), and define a lax functor $F: C \to \textbf{Span}$. For simple minded people like me this means that I'm not sending identities to identities and compositions to compositions, but I have morphisms of spans relating $Fid_x$ with $id_{FX}$, and $F(f;g)$ with $Ff ; Fg$, such that some diagrams commute. I've already checked that in my case all this stuff goes through.

Now, I've also some natural transformations that - I believe - show that the monoidal structure of $C$ is laxly preserved by $F$. Again, in simple words this means that I have a span $F(A \otimes B) \to FA \times FB$ for each $A,B$, and a span $\{*\} \to FI$ where $I$ is the monoidal unit of $C$, and that the usual commutative squares for lax monoidal functors commute up to a morphism of spans (which is not an iso, in my case).

My question is: What are the precise coherence conditions I've to check to be able to say that I have some sort of lax monoidal lax functor between $C$ and $\textbf{Span}$?
In other words, what is the right definition for a lax functor between monoidal bicategories that is also lax on the monoidal structure?

I've thought that maybe I could consider $\textbf{Span}$ as a tricategory with just one object, and try to prove that I have a lax 3-functor between $C$ and $\textbf{Span}$, but I'm really a noob in higher category theory and I don't know if this is the right way of doing things.

John Baez (Oct 28 2020 at 16:46):

So it sounds like you need to take the definition of a monoidal functor between monoidal bicategories and "laxify" it by dropping the requirement that various things are isomorphisms or equivalences.

John Baez (Oct 28 2020 at 16:47):

If so, you need to get ahold of the definition of monoidal functor between monoidal bicategories.

John Baez (Oct 28 2020 at 16:49):

It's buried in here:

https://arxiv.org/abs/1112.1000

starting in section 2.1. The history and references here are very useful.

This paper has more than you want, since it also explains braided, sylleptic and symmetric bicategories and the appropriate functors between those, and also the appropriate natural transformations between those.

John Baez (Oct 28 2020 at 16:54):

However, the good news: the stuff relevant to monoidal bicategories is labeled with a "club" (as in a deck of cards). So you can go to Section 2.3 and start reading, paying attention only to the stuff labeled with clubs.

John Baez (Oct 28 2020 at 16:57):

Definition 2.5 is what you actually want: the definition of "symmetric monoidal homomorphism", but only keeping the stuff labeled with clubs.

John Baez (Oct 28 2020 at 16:58):

Now I see something really annoying: he "outsources" the coherence laws, meaning he gives references to specific equations in other people's papers, instead of writing down the coherence laws. :hurt:

Reid Barton (Oct 28 2020 at 17:00):

To be fair, a reference to an equation takes a lot less than a whole page to write down.

John Baez (Oct 28 2020 at 17:00):

It would have been a much more valuable reference if he included all the diagrams.

Reid Barton (Oct 28 2020 at 17:00):

While the same is probably not true of the equations themselves.

John Baez (Oct 28 2020 at 17:01):

This paper - actually Chris Schommer-Pries' thesis - is already 317 pages long, so 20 or 30 more pages wouldn't have hurt.

Fabrizio Genovese (Oct 28 2020 at 19:44):

Yes, I thought that Schommer-Pries thesis was the right place to start from. Still, probably using lax-3-functors is going to be a bit easier?

John Baez (Oct 28 2020 at 20:10):

I guess it will ultimately be the same, since Gordon Power and Street (GPS) defined maps between monoidal bicategories to be maps between one-object tricategories. There are some nuances in this general vicinity, but I don't think they kick in yet here. I just remembered a good reference:

• Eugenia Cheng and Nick Gurski, The periodic table of n-categories II: degenerate tricategories .

They describe a tricategory MonBicat, among other things. And they carefully analyze all the nuances that I'm worrying about.

The title on the heading of each page has an amusing typo....

Fabrizio Genovese (Oct 28 2020 at 21:32):

This is really helpful, thanks!