Polynomial functors and opetopes

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.

LaTeX, 54 pages, 75 texdraw figures. Accompanying opetope scripts in Tcl hidden in tex source after \end{document} for the sake of archival -- also available from http://mat.uab.cat/~kock/cat/zoom.html . v2: substantial expository improvements, following the advice from the referees. Final version, to appear in Adv. Math

References in corpus (4)

• Polynomial functors and polynomial monads
• The category of opetopes and the category of opetopic sets
• Computads and Multitopic Sets
• Polynomial functors and trees

Cited by in corpus (19)

• Polynomial functors and polynomial monads
• Homotopy theory for algebras over polynomial monads
• Groupoids and Faa di Bruno formulae for Green functions in bialgebras of trees
• Koszul duality for operadic categories
• Cacti and filtered distributive laws
• $\infty$-Operads as Analytic Monads
• Polynomial functors and combinatorial Dyson-Schwinger equations
• Operadic categories as a natural environment for Koszul duality
• Categorification of Hopf algebras of rooted trees
• The incidence comodule bialgebra of the Baez-Dolan construction
• A Cartesian Bicategory of Polynomial Functors in Homotopy Type Theory
• Polynomial functors and trees
• Multitopes are the same as principal ordered face structures
• Polynomial and Analytic Functors and Monads, revisited
• Combinatorial structure of type dependency
• The Web Monoid and Opetopic Sets
• Higher dimensional algebras via colored PROPs
• From dependent type theory to higher algebraic structures
• Types are Internal $\infty$-Groupoids