### Bar Constructions and Combinatorics of Polyhedra for n-Categories

#### Posted by John Baez

Samuel Vidal has kindly LaTeXed some notes by Todd Trimble:

Todd wrote these around 1999, as far as I know. I’ve always enjoyed them; they give a clearer introduction to the bar construction than any I’ve seen, and they also suggest a number of fascinating directions for research on the relation between higher categorical structures and polyhedra.

Here’s the basic idea:

In these notes, we give some combinatorial techniques for constructing various polyhedra which appear to be intimately connected with the geometry of weak n-categories. These include

- Associahedra (and “monoidahedra”; see below);
- Permutoassociahedra, and the cellular structure of Fulton-MacPherson compactifications of moduli spaces
- “Functoriahedra” (related to the $A_n$ maps of Stasheff)
Our basic techniques use derivations on operads and bar constructions. In part A, we introduce derivations, which should be regarded as “boundary operators” on (set-valued) operads satisfying a Leibniz rule. Such derivations can be used to construct poset-valued operads; taking nerves, one gets polyhedral operads. In this way we reconstruct the associahedra and the Fulton-MacPherson compactifications.

By the way, Niles Johnson recently added better pictures of the 3d associahedron to Wikicommons:

## Re: Bar Constructions and Combinatorics of Polyhedra for n-Categories

They were originally in the form of an email to John. IIRC, most of the ideas were in my head around 1997 or so. I no longer recall what prompted me to send that email.

Yes, it was indeed very kind of Samuel to TeX up those notes. I may be adding some more material to the TeX file he shared.

I believe Stefan Forcey, who has been a patron of the Café from time to time, has pursued similar themes – but unlike me, he’s published on such things. The associahedra are of course very well-known, but the “functoriahedra” (as I called them), describing the polyhedral shapes for weak $n$-functors – not as much. I believe Stefan has written about those as well. I never did work out what should be the geometry of higher transfors.

I’m happy to discuss these things with whoever may be interested.