Combinatorial homotopy theory for operads

Abstract

We introduce an explicit combinatorial characterization of the minimal model ${\mathcal{O}}_{\infty }$ of the coloured operad $\mathcal{O}$ encoding non-symmetric operads. In our description of ${\mathcal{O}}_{\infty }$, 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 $\mathcal{O}$, as well as of the minimal model of the coloured operad $\mathcal{C}$ encoding non-symmetric cyclic operads.

36 pages