Symmetries of genus zero modular operad

TL;DR

This paper introduces a categorical framework for understanding symmetries of genus zero modular operads, combining modern persistence homology techniques with classical groupoid formalism, revealing new insights into the associated symmetry groups.

Contribution

It presents a novel categorical approach to symmetries of genus zero modular operads, linking persistence homology with classical groupoid methods and identifying a new form of the profinite Grothendieck-Teichmüller group.

Findings

New categorical framework for operad symmetries

Identification of a 'poset in groupoids' as a symmetry structure

Connection to a new avatar of the profinite Grothendieck-Teichmüller group

Abstract

In this article combining survey and certain research results, we introduce a categorical framework for description of symmetries of genus zero modular operad. This description merges the techniques of recent "persistence homology" studies and the classical formalism of groupoids. We show that the contravariant "poset in groupoids" embodying these symmetries, provides a new avatar of profinite Grothendieck-Teichm\"uller group acting upon this operad but seemingly not related with representations of the Galois group of all algebraic numbers.