solifine

The objects of the simplicial category 𝚫 are [n] = {0 < 1 < ⋯ < n}, for n≥0, and instead of writing Δⁿ for the representable simplicial set hom(-,[n]), let's just write [n]. (Mostly so we can actually write [mᵢ] on twitter!) ⌊03⌋

What are the operad structure maps? The tensor unit for ⨿ is ∅, so we don't have any choice about the "unit" ∅ → Δ¹ = [1] in our operad. ⌊04⌋

To write the operadic multiplications γ: [r] ⨿ ([m₁] ⨿ ⋯ ⨿ [mᵣ]) → [Σmᵢ], maybe it's good to write elements like (k,r) and (k,mᵢ) to specify which component we're starting in. ⌊05⌋

(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⌋

For each simplicial set X, we have produced an X(1)-colored cooperad [in (Set,×,∗)]. ⌊14⌋

If X happens to be "2-Segal" then this cooperad is "invertible". This means we can reverse the structure maps (which are bijections), and so our simplicial set is also an operad. ⌊15⌋