Combinatorics and simplicial groupoids

TL;DR

This paper explores the connections between combinatorics, simplicial groupoids, and algebraic structures like the Faa di Bruno and plethystic bialgebras, providing an objective, category-theoretic construction.

Contribution

It introduces a novel simplicial groupoid-based framework to construct important combinatorial bialgebras objectively.

Findings

Constructs Faa di Bruno bialgebra using simplicial groupoids

Provides a categorical perspective on combinatorial species

Links incidence algebras with simplicial groupoid theory

Abstract

This expository paper starts with a brief survey on the relation between partitions and surjections of sets, and then gives a quick introduction to the theories of incidence algebras, Segal groupoids and combinatorial species. The aim is to explain an objective construction, in terms of simplicial groupoids, of both the Faa di Bruno bialgebra and the plethystic bialgebra.