solifine

(ofc you should be careful, bc in real life you could have, like, m₁=m₂=r=5 and then you'll get confused) ⌊06⌋

Are you satisfied that the standard simplices form an operad? If so, let X be a simplicial set. We know that hom(A⨿B,X) = hom(A,X)×hom(B,X) and X(n) = hom(Δⁿ,X). (here, hom means "simplicial set morphisms", while earlier it meant morphisms in 𝚫) ⌊09⌋

It follows that the maps γ from above give functions X(Σmᵢ) → X(r)×(X(m₁)×⋯×X(mᵣ)) and the unique map ∅ → [1] gives X(1) → hom(∅,X) = ∗. ⌊10⌋

Because the maps γ were not injective, the cocomposition map X(Σmᵢ) → X(r)×(X(m₁)×⋯×X(mᵣ)) lands in a smaller subset. ⌊11⌋

More specifically, this is an X(1)-colored cooperad. The "output color" of an element x ∊ X(n) is found using the endpoint preserving function [1] → [n] in 𝚫 (that is, 0↦0, 1↦n). ⌊12⌋

For 1≤j≤n, the jth "input color" of x∊X(n) is given by the interval inclusion [1] → [n] in 𝚫 that sends 0 to j-1 and 1 to j. ⌊13⌋