An infinity operad of normalized cacti

A cell decomposition of the Fulton MacPherson operad

“…However, the spaces $${\mathcal {C}}_k$$ themselves do not form a topological operad. They only form an operad up to higher homotopies [7]. The composition of cacti $$c\ \circ _i\ d$$, roughly speaking, glues $$d$$ into the $$i$$th lobe of $$c$$, by identifying the global base point of $$d$$ and the local base point of the $$i$$th lobe of $$c$$.…”