### Trimble’s Definition of Tetracategory

#### Posted by John Baez

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

- Todd Trimble, Notes on tetracategories.

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…

## 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”?