solifine

Every simplicial set X is a (nonsymmetric) cooperad, whose n-ary cooperations are precisely the n-simplices of X. Why? Where do the cooperadic cocompositions come from? ⌊01⌋

First off, the standard simplices Δⁿ (as n≥0 varies) form an operad in simplicial sets. Wait wait, simplicial sets with COPRODUCT as the monoidal product! How does this work? ⌊02⌋

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⌋