## October 29, 2018

### 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

1. Associahedra (and “monoidahedra”; see below);
2. Permutoassociahedra, and the cellular structure of Fulton-MacPherson compactifications of moduli spaces
3. “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:

Posted at October 29, 2018 3:25 PM UTC

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### 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.

Posted by: Todd Trimble on October 30, 2018 1:42 AM | Permalink | Reply to this

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

I have worked a bit on coherence of transfors in low dimensions, and would be interested in looking into it more. My setting was a bit different though so I am not sure if I understand everything you are doing.

From what I understand, the difficulty lies in finding an ‘operad of transfors’, is that right? Are the “free bimodules”-monads of Example 11 and 12 actually operads?

Posted by: Maxime LUCAS on November 6, 2018 1:19 PM | Permalink | Reply to this

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

The geometry of higher transfors is pretty well-understood in the strict case nowadays. Do you think it’s actually possible to give polyhedral coherence diagrams for the pseudo version?

Posted by: Harry Gindi on December 25, 2018 11:51 AM | Permalink | Reply to this

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

Thanks for the mention Todd! I only now had the chance to reply. The “functoriahedra” were first described by Jim Stasheff in 1970, and given the name multiplihedra soon after. I have a little encylopedia entry about them, with links to some categorical uses—like trihomomorphism axioms. I’ll stop by later with some links about higher transfors, which got some attention in Steve Lack’s thesis.

Posted by: stefan on January 10, 2019 10:45 PM | Permalink | Reply to this

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

There’s a slight error in the second half of the diagram. The value just NE of center should be a(b((cd)e)), instead of duplicating the a((b(cd))e) just to its lower right.

Posted by: Craig Helfgott on November 5, 2018 4:59 PM | Permalink | Reply to this

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

Thanks! I’ve informed the creator of the diagram, Niles Johnson.

Posted by: John Baez on November 5, 2018 6:36 PM | Permalink | Reply to this

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

Thanks for noticing! Fixing it now :) As it turns out, I noticed and fixed an error in the other diagram just this morning – too many parentheses in (a((bc)d)))e.

Posted by: Niles Johnson on November 5, 2018 8:37 PM | Permalink | Reply to this

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