Moduli Operad over F1

In this paper we answer a question raised in [25], Sec. 4, by showing that the genus zero moduli operad $\{{\bar{M}}_{0,n+1}\}$ can be endowed with natural descent data that allow it to be considered as the lift to $\mathrm{S}\mathrm{p}\mathrm{e}\mathrm{c}\mathbb{Z}$ of an operad over ${\text{F}}_1$. The relevant descent data are based on a notion of constructible sets and constructible functions over ${\text{F}}_1$, which describes suitable differences of torifications with a positivity condition on the class in the Grothendieck ring. More generally, we do the same for the operads $\{T_{d,n+1}\}$ (whose components were) introduced in [5]. Finally, we describe a blueprint structure on $\{{\bar{M}}_{0,n}\}$ and we discuss from this perspective the genus zero boundary modular operad $\{{\bar{M}}_{g,n+1}^0\}$.