Higher Hopf algebrasMPIM

by Ralph Kaufmann (Purdue/MPIM)

Tuesday Aug 11, 2026, 11:00 AM → 12:30 PM Europe/Berlin

MPIM, Vivatsgasse, 7 - Seminar Room (Max Planck Institute for Mathematics)

Abstract homotopy seminar

Together with Galvez-Carilllo and Tonks we established that every simplicial set yields a bi-algebra, whose cubical structure can be understood via the category of simplicial strings, aka. necklaces, using Joyal duality. The reason for the existence is the fact that simplices form an operad and after dualization give a co-operad with multiplication which is the correct input for the GCKT machine.
We aim to understand and to construct strict higher categorical versions of this. The first ingredients are Street’s orientals and a monoidal product explained by Ozornova and Rovelli for higher categories defined by polytopes with an initial and a final vertex. Note that even the classical path space argument is actually a transition from a 2 to a 1 category. Following the Kapranov-Voevodsky dream the next steps are cubes and permutahedra. For the latter there is is the Manin-Schechtman theory of higher Bruhat orders and the relatively new operad with multiplication structure on them defined by Koshevoy and Schechtman. This project is joint with Ozornova, Koshevoy and Schechtman.