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arXiv · 0706.1033

Polynomial functors and opetopes

Joachim Kock·André Joyal·Michael Batanin·Jean-François Mascari
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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.

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Joachim Kock, André Joyal, Michael Batanin, Jean-François Mascari. 2010-02-21. Polynomial functors and opetopes. https://doi.org/10.1016/j.aim.2010.02.012

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