Decomposition spaces: maps, convexity and Crapo complementation formula (Imma Galvez)

Orateur : Imma Galvez

Lieu : salle des Séminaires M3

Résumé :
Decomposition spaces, also known as 2-Segal spaces, are simplicial infinity-groupoids that send active-by-inert pushouts in the simplicial category to homotopy pullbacks. They give rise to coalgebra and algebra structures whose homotopy cardinalities are classical bialgebras.

To obtain (co)algebra homomorphisms, we consider maps given by spans of simplicial infinity-groupoids that fulfil certain conditions, called CULF on one leg and IKEO on the other.  To formulate and understand these conditions we will go through several equivalent definitions of decomposition spaces. Finally, we will develop a concept of convexity and we will prove the classical complementation formula of Crapo in this context.

Joint work with Joachim Kock and Andrew Tonks.
(https://arxiv.org/abs/2409.03742)