A simplicial groupoid for plethysm

TL;DR

This paper introduces a combinatorial model for plethysm by representing its dual bialgebra as the homotopy cardinality of an incidence bialgebra derived from a simplicial groupoid, linking algebraic and topological structures.

Contribution

It provides a novel combinatorial and homotopical framework for understanding plethysm through simplicial groupoids and incidence bialgebras.

Findings

Realizes the dual of plethystic substitution as a homotopy cardinality.

Constructs an explicit simplicial groupoid from surjections.

Connects plethysm with topological and combinatorial models.

Abstract

We give a simple combinatorial model for plethysm. Precisely, the bialgebra dual to plethystic substitution is realised as the homotopy cardinality of the incidence bialgebra of an explicit simplicial groupoid, obtained from surjections by a construction reminiscent of the Waldhausen S and the Quillen Q-construction.