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 ${\rm Spec}\,\Z$ of an operad over $\F_1$. The relevant descent data are based on a notion of constructible sets and constructible functions over $\F_1$, 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 $\{\bar{M}_{g,n+1}^0\}$.