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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 nA_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 nn-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

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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