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John Baez (Nov 27 2020 at 21:41):My former student @Alex Hoffnung may be one of the few people to have proved something is a monoidal tricategory:
I've always been sad that this paper was never published. But I just heard some news about this: he's getting back to work on getting it published!
John Baez (Nov 27 2020 at 21:41):The paper is 137 pages long, but he's going to trim it down and refer to the long version for some details of the proofs, which involve a lot of complicated diagrams.
Jules Hedges (Nov 27 2020 at 21:53):Monoidal tricategories is an objectively correct use of our new :mind-blown: emoji: 
Fabrizio Genovese (Nov 28 2020 at 01:03):Lol, every time I stumble in a tricategory (not necessarily monoidal) I give up on whatever idea I'm thinking about, thinking that it's not worth the effort. I really envy the level of determination you need to prove that something is a monoidal tricategory. Kudos.
John Baez (Nov 28 2020 at 04:36):It's impressive indeed! Someday a better approach will make all of this take less work...
Rune Haugseng (Nov 28 2020 at 17:27):Maybe this is controversial, but don't we already have such a better approach, in the form of the homotopical version of higher categories? Certainly (symmetric) monoidal -categories are pretty easy to define, as are -categories of (iterated) spans...
John Baez (Nov 28 2020 at 18:55):Yes, for those like you who can handle -categories there is already a better approach. As William Gibson said, "the future is already here, it's just not evenly distributed". I have not been keeping up with these issues anymore.
John Baez (Nov 28 2020 at 18:57):At some point I decided I should stick to -categories with , and focus on applying them to new subjects, like electrical engineering and chemistry.
John Baez (Nov 28 2020 at 19:05):So I guess it would be helpful if those who understand -categories to write some easy expository articles explaining them. Or maybe these articles exist - I'm not looking for them - in which case advertising them might be good.
John Baez (Nov 28 2020 at 19:06):If -categories ever seemed "easy" to me, I can imagine wanting to use them.
Fabrizio Genovese (Nov 28 2020 at 21:19):I may be wrong, but I feel that approaching such problem from an infinity categories point of view is useful if you have to prove things, less useful when you have to crunch out numbers. Lately I'm playing with particular kinds of spans seen as a lax monoidal bicategory. I need to prove the existence of some lax symmetric monoidal lax functor. In the end, I'm doing applied stuff, so I need to be able to build such spans "manually" and be sure that all the structural maps do stuff that fits my examples. I have the feeling that yes, I can prove such a lax monoidal lax functor exists using more abstract methods but it will still a lot of work to explicitly say what all the structural maps do afterwards...
John Baez (Nov 28 2020 at 21:47):It will be very interesting - sociologically at least - when the -category people and the applied category people start seriously talking to each other. The applied people will push for structures that can easily be implemented on the computer without too much data; the -people will push for elegance and generality... and if they don't get into a big fight and stop talking to each other, some marvelous new techniques may emerge!
John Baez (Nov 28 2020 at 21:51):For the last ten years I've only been focused on getting 1-category theorists to start seriously talking to applied mathematicians and engineers. It's really important for category theory to come down from the sky and "touch ground" - that is, accomplish things that even nonmathematicians can recognize as worthwhile. I want it to happen not just in computer science (where it arguably already has) but in subjects like electrical engineering, logistics, chemistry, ecology and so on.
John Baez (Nov 28 2020 at 21:52):It's like when a small stroke of lightning first hits ground: there's a powerful "return stroke" that goes back up, once a conductive path has been formed.
John Baez (Nov 28 2020 at 21:54):Here's a video of a "return stroke":
https://www.youtube.com/watch?v=QtwdQrMwedQ
Right now we're in the phase where little tentative strokes are coming down, branching in different directions, trying to see how to get down to earth...
Jules Hedges (Nov 28 2020 at 21:56):Presumably there is some for which a lazy person would prefer the -approach to the -approach, just because it's technically easier. Is there some consensus about the value of where that happens? I'd guess it's somewhere around 4...?
Jules Hedges (Nov 28 2020 at 21:59):On the other hand I expect the maximum dimension that ever comes up in applications to have a hard upper bound given by what the gods allow us to visualise in our 3+1 dimensional universe. I think that means monoidal tricategories is the biggest thing you could visualise, with movies of surfaces
John Baez (Nov 28 2020 at 22:01):I think that value of is somewhere around 3 or 4.
John Baez (Nov 28 2020 at 22:02):If we ever come up with a really n-categorical theory of physics in 4d spacetime, it may involve a tetracategory, e.g. a monoidal tricategory, so interestingly this is right at that borderline - perhaps no coincidence since paper needs to be at most 2-dimensonal in 4d spacetime, and that sets the limitations on our syntax.
John Baez (Nov 28 2020 at 22:05):On the other hand if superstring theory is right and the universe is fundamentally 10- or 11-dimensional, an n-categorical theory of physics will probably need some sort of more abstract approach to understand, since we may need 10- or 11-categories.
Fabrizio Genovese (Nov 28 2020 at 22:18):Jules Hedges said:
On the other hand I expect the maximum dimension that ever comes up in applications to have a hard upper bound given by what the gods allow us to visualise in our 3+1 dimensional universe. I think that means monoidal tricategories is the biggest thing you could visualise, with movies of surfaces
Not sure about that. A glorious future could be one where your intuition is just 1-categorical (or a bit more), and when things do not commute as they should there's a computer that takes care of all the higher $$n$$s for you to make it work.
Fabrizio Genovese (Nov 28 2020 at 22:18):This is, very handwavily, what HoTT seems to promise, at least for .
John Baez (Nov 28 2020 at 22:21):So far computers can't even handle a double dollar sign touching a character: $$n$$s versus s.
Fabrizio Genovese (Nov 28 2020 at 22:22)::smile:
Matteo Capucci (he/him) (Nov 28 2020 at 23:06):Lately I was wondering if such a reconciliation might be really easy working in HoTT? It's a very hands-on, computationally prone approach to higher-stuff, though it misses 'directedness', i.e. types are -groupoids aka -categories.
However, is hard to ignore the heavy emphasis HoTT puts on the concept of '-equivalence'! And the heavy applicability of it to homotopy theory, which is incidentally shared by (at least a strain of) higher category theory.
In a sense, HoTT and quasi-groupoids ($\infty$-groupoids à là Lurie-Joyal) are in the same relationship cubical and simplicial sets are, if I am not mistaken (one may want to replace HoTT with Cubical Type Theory though)
Eric Forgy (Nov 28 2020 at 23:11):(one may want to replace HoTT with Cubical Type Theory though)
Or "diamond" type theory :blush:
Matteo Capucci (he/him) (Nov 28 2020 at 23:16):Eric Forgy said:
(one may want to replace HoTT with Cubical Type Theory though)
Or "diamond" type theory :blush:
Is this a thing or just a meant to be suggestive? :laughing: Googling 'diamond type theory' I found this book and it took me some time to understand is probably crackpot 'mathematics'
Notification Bot (Nov 30 2020 at 10:46):This topic was moved here from #general > Spans in 2-categories by Matteo Capucci
John Baez (Dec 09 2020 at 21:36):It turns out both Power and Street recommend that he resubmit his paper more or less "as is" to TAC, just cutting out his review of known material about pseudolimits. Power in particular was not worried about the length of the paper.
David Michael Roberts (Dec 15 2020 at 05:37):Hurray for electronic journals with no page limits!
John Baez (Dec 15 2020 at 21:36)::+1: