# Alex Cebrian: A simplicial groupoid for plethysm

**Time: **
Tue 2019-05-07 13.15 - 15.00

**Lecturer: **
Alex Cebrian (UA Barcelona)

**Location: ** Room 3418, Lindstedtsvägen 25, 4th floor, Department of Mathematics KTH

ABSTRACT: Plethysm is a substitution operation in the ring of formal power series in infinitely many variables. We will give a combinatorial model for this operation in terms of simplicial groupoids. More precisely, we will define the plethystic bialgebra, whose comultiplication is dual to plethystic substitution, and realize it from an explicit Segal space through the standard constructions of incidence coalgebras and homotopy cardinality of groupoids, which will also be explained. Many combinatorial coalgebras can be obtained using these tools. For instance, we will also see the analogous result for substitution of one variable power series, which recovers the

Fa\`a di Bruno bialgebra from the fat nerve of the category of surjections.

Fa\`a di Bruno bialgebra from the fat nerve of the category of surjections.