Half-balanced braided monoidal categories and Teichmueller groupoids in genus zero

TL;DR

This paper introduces new categorical structures called half-balanced braided monoidal categories and their contractions, providing an explicit description of Galois group actions on genus zero Teichmueller groupoids, linking to existing work by Schneps.

Contribution

It defines half-balanced braided monoidal categories and their contractions, and relates these to Galois actions on Teichmueller groupoids in genus zero, offering a new categorical perspective.

Findings

Explicit description of Galois group action on Teichmueller groupoids

Equivalence between prounipotent and graded actions

Connection to Schneps' results on genus zero

Abstract

We introduce the notions of a half-balanced braided monoidal category and of its contraction. These notions give rise to an explicit description of the action of the Galois group of QQ on Teichmueller groupoids in genus 0, equivalent to that of L. Schneps. We also show that a prounipotent version of this action is equivalent to a graded action.