Hopf monoids in the category of species

TL;DR

This paper introduces the theory of Hopf monoids within the category of species, illustrating their structure through combinatorial examples and exploring fundamental algebraic properties and theorems.

Contribution

It provides a comprehensive, self-contained framework for Hopf monoids in species, highlighting the role of the Tits algebra and classical algebraic theorems in this context.

Findings

Identification of the Tits algebra as a universal operation

Examples of combinatorial structures forming Hopf monoids

Discussion of classical algebraic theorems in the Hopf monoid setting

Abstract

A Hopf monoid (in Joyal's category of species) is an algebraic structure akin to that of a Hopf algebra. We provide a self-contained introduction to the theory of Hopf monoids in the category of species. Combinatorial structures which compose and decompose give rise to Hopf monoids. We study several examples of this nature. We emphasize the central role played in the theory by the Tits algebra of set compositions. Its product is tightly knit with the Hopf monoid axioms, and its elements constitute universal operations on connected Hopf monoids. We study analogues of the classical Eulerian and Dynkin idempotents and discuss the Poincare-Birkhoff-Witt and Cartier-Milnor-Moore theorems for Hopf monoids.