June 16, 2007

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

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