### Opetopes as Trees

#### Posted by David Corfield

How did this one slip by? Four important players in the saga of $n$-categories, Joachim Kock, André Joyal, Michael Batanin and Jean-François Mascari have combined to write Polynomial functors and opetopes:

We give an elementary and direct combinatorial definition of opetopes in terms of trees, well-suited for graphical manipulation (e.g. drawings of opetopes of any dimension and basic operations like sources, target, and composition); a substantial part of the paper is constituted by drawings and example computations. To relate our definition to the classical definition, we recast the Baez-Dolan slice construction for operads in terms of polynomial monads: our opetopes appear naturally as types for polynomial monads obtained by iterating the Baez-Dolan construction, starting with the trivial monad. Finally we observe a suspension operation for opetopes, and define a notion of stable opetopes. Stable opetopes form a least fixpoint for the Baez-Dolan construction. The calculus of opetopes is also well-suited for machine implementation: in an appendix we show how to represent opetopes in XML, and manipulate them with simple Tcl scripts.

I see in the bibliography a reference to Notes on Polynomial Functors by Kock.

Posted at June 16, 2007 4:51 PM UTC
## Re: Opetopes as Trees

I printed that one the day it hit the arXiv. I knew you and your compatriots would orbit around it, giving annotated views of the scenery. Parts of it make sense straight through. Much is deep, and requires illumination, to those such as I, jetting about the tree-coral and brain-coral in squid-inky darkness.