McClure-Smith cosimplicial machinery and the cacti operad

(eng) McClure and Smith constructed a functor that sends a topological multiplicative operad O to an E2 algebra TotO•. They define in fact an operad D2 (acting on the totalization TotO•) weakly equivalent to the little 2-disks operad. On the other hand, Salvatore showed that D2 is isomorphic to the cacti operad MS, which has a nice geometric description. He also built a geometric action of MS on TotO•. In this paper we detail the McClure–Smith action and the cacti action. Our main result says that they are compatible in the sense that some squares must commute.