  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.
