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 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 -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.