Symmetries of genus zero modular operad

Integrability, Quantization, and Geometry, Vol. II., AMS Proceedings of Symposia in Pure Mathematics, 2021 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.