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October 5, 2006

Trimble’s Definition of Tetracategory

Posted by John Baez

Trimble’s legendary lost definition of “tetracategory”is now available here:

As some of you know, the definition of “category” takes a few lines, while the definition of “bicategory” takes a couple pages and the definition of “tricategory” takes quite a few. This makes a direct nuts-and-bolts definition of “tetracategory” a formidable challenge. Trimble tackled this back in 1995. His tale begins as follows:

In 1995, at Ross Street’s request, I gave a very explicit description of weak 4-categories, or tetracategories as I called them then, in terms of nuts-and-bolts pasting diagrams, taking advantage of methods I was trying to develop then into a working definition of weak n-category. Over the years various people have expressed interest in seeing what these diagrams look like – for a while they achieved a certain notoriety among the few people who have actually laid eyes on them (Ross Street and John Power may still have copies of my diagrams, and on occasion have pulled them out for visitors to look at, mostly for entertainment I think).

In Trimble’s definition, the star of the show is the 4-dimensional associahedron which generalizes the “pentagon identity” familiar from bicategories. His 16-page diagram of this 4d associahedron survives from 1995. But, there are also 30 pages of diagrams describing coherence conditions for unit laws, and Trimble lost his copies of these.

Now he has redrawn them, and kindly made them available to the world, along with some notes explaining the idea behind them. This material will not be to everyone’s taste, but as he notes,

Despite their notorious complexity, there seems to be some interest in having these diagrams publicly available…

Posted at October 5, 2006 3:08 AM UTC

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Re: Trimble’s Definition of Tetracategory

How’s this for an idea? Given Todd’s willingness to write notes, and your concern (e.g., here) that Jim Dolan’s ideas are not being written up, couldn’t Todd be prevailed upon to jot down some notes about the discussions you mentioned they had together “on categorifying everything associated to simple Lie groups and quantum groups”?

Posted by: David Corfield on October 5, 2006 8:55 AM | Permalink | Reply to this

Re: Trimble’s Definition of Tetracategory

Todd has already written some notes on this subject, and he plans to write more. As material becomes available, it will show up on my webpage devoted to his work.

Posted by: John Baez on October 5, 2006 3:34 PM | Permalink | Reply to this

Re: Trimble’s Definition of Tetracategory

Can anyone map the table at the end of this post onto the Baez-Dolan k-monoidal n-category table? ‘Resultoassociahedron’ gets 1 hit on Google, to that post.

Posted by: David Corfield on October 5, 2006 9:16 AM | Permalink | Reply to this

Re: Trimble’s Definition of Tetracategory

How does the resultassociahedron compare
to the reultohedron??

CTRC Seminar Abstract: How braidings the post-modern way give rise …
… C tensor G C -> C. Spelling out functoriality of R in both variables one gets an equation of the shape of Kapranov and Voevodsky’s resultohedron N2,2. … - 7k - Cached - Similar pages

[PDF] arXiv:math.CT/0207281 v2 16 Sep 2003
File Format: PDF/Adobe Acrobat - View as HTML
is the resultohedron N. pq. [22, 17]. We finish this example by presenting a picture for multiplication in P (or G if. you like). The reader might find this …

Posted by: jim stasheff on October 13, 2006 4:17 PM | Permalink | Reply to this

Re: Trimble’s Definition of Tetracategory

Speaking of lost stuff, I have heard in quite a few places mentions of Chevalley’s lost paper on limits (it’s in MacLane’s book, and definitely a few other places I can’t remember as well); any of you guys know anything about it? (Oh, much thanks for the chat on Doctrines guys; I remember trying to read Lawvere’s paper on them a while ago and just ending up feeling rather confused).

Posted by: Stephen Lavelle on October 19, 2006 12:51 PM | Permalink | Reply to this

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