Combinatorial homotopy theory for operads

We introduce an explicit combinatorial characterization of the minimal model \({\cal O}_{\infty}\) of the coloured operad \({\cal O}\) encoding non-symmetric operads. In our description of \({\cal O}_{\infty}\), the spaces of operations are defined in terms of hypergraph polytopes and the composition structure generalizes the one of the \(A_{\infty}\)-operad. As further generalizations of this construction, we present a combinatorial description of the \(W\)-construction applied on \({\cal O}\), as well as of the minimal model of the coloured operad \({\cal C}\) encoding non-symmetric cyclic operads.