Moduli Operad over F 1 Yuri

In this paper we answer a question raised in [25], Sec. 4, by showing that the genus zero moduli operad {M0,n+1} can be endowed with natural descent data that allow it to be considered as the lift to SpecZ of an operad over F1. The relevant descent data are based on a notion of constructible sets and constructible functions over F1, 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 {Td,n+1} (whose components were) introduced in [5]. Finally, we describe a blueprint structure on {M0,n} and we discuss from this perspective the genus zero boundary modular operad {M0g,n+1}.