Opetopic shapes, combinatorially
Pierre-Louis Curien
CNRS and Université Paris Cité
Abstract:
(dedicated to the memory of Marek Zawadowski, both a friend and a very smart scientist)
Opetopes and opetopic sets were introduced by Baez and Dolan at the end of the previous millenium as a framework for dealing with higher structures and higher coherences. They are many-to-one shapes in all dimensions and can be represented in different ways: abstractly through the iterated so-called plus or slice construction on polynomial monads, and more concretely as zoom complexes (Kock-Joyal-Batanin-Mascari), or as some suitable oriented posets of faces as in the works of Zawadowski. In a series of works, Louise Leclerc, and Louise Leclerc and myself have laid down precise isomorphisms between these combinatorial presentations, adding one in the picture: opetopes as epiphytes, which are recursively defined as trees whose nodes and edges are decorated with codimension 1 and 2 epiphytes, respectively, and are arguably most faithful to the original abstract setting. The talk will offer a (gentle) journey through these (quite involved) combinatorial descriptions (epiphytes, zoom complexes, oriented face structures).
Joint work with Louise Leclerc.