Polynomial functors and opetopes

Details Polynomial functors and opetopes Polynomial functors and opetopes

2007-06-07

arxiv.org/abs/0706.1033v2

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

Mathematics

Quantum Algebra

LaTeX, 54 pages, 75 texdraw figures. Accompanying opetope scripts in Tcl hidden in tex source after \end{document} for the sak

Scientific paper

10.1016/j.aim.2010.02.012

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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