the operad structure of $\overline{M_{0,n+1}}({\mathbb{R}})$

The category of representations over a quantum group $U_q(g)$ form a braided tensor category that produces an action of the (pure) braid groups on tensor products. Respectively, the category of crystals (which is a limit for q tends to zero) form a coboundary category together with an action of (pure) cacti group on tensor products. The little discs operad is an operad whose space of $n$-ary operations is the Eilenberg-Maclein space of the pure braid groups with $n$ braids. Correspondingly, the real locus of the Deligne-Mumford compactification of the moduli space of stable rational curves with marked points assemble an operad of the Eilenberg-Maclein spaces of pure cacti groups. I will present the detailed description of the latter operad as well as its deformation theory and relationships with the little discs operad, graph complexes and Grothendieck-Teichmuller Lie algebra