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# Polynomial functors and opetopes

By Joachim Kock, André Joyal, Michael Batanin, and Jean-François
Mascari

*Adv. Math.* **224**
(2010) 2690-2737.

ArXiv:0706.1033

#### Abstract

We give an elementary and direct combinatorial definition of opetopes in terms
of trees, well-suited for graphical manipulation and explicit computation. 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. We show that our
notion of opetope agrees with Leinster's. Next 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. A final
section is devoted to example computations, and indicates also how the
calculus of opetopes is well-suited for machine implementation.

Download the paper

See examples of opetopes in high dimension

Read the documentation for the opetope scripts

Download the package of scripts and example XML files

Try
Eric
Finster's online interactive opetope
explorer

Last updated: 2011-09-08 by
Joachim Kock.