following [@BM] and [@BM-colors], we call a map $f: {{\cal O}}\to  {{\widetilde{{\mathcal O}}}}$ of colored operads a [*fibration*]{} if for every collection of colors $c_1, \dots, c_k, c$ the map $$f : {{\cal O}}(c_1, \dots, c_k ; c) \to  {{\widetilde{{\mathcal O}}}}(c_1, \dots, c_k ; c)$$ is surjective. We say that $f: {{\cal O}}\to  {{\widetilde{{\mathcal O}}}}$ is an [*acyclic fibration*]{} is $f$ is a quasi-isomorphism and a fibration. A map $f: {{\cal O}}\to  {{\widetilde{{\mathcal O}}}}$ is called a [*cofibration*]{} if $f$ satisfies the left lifting property with respect to all acyclic fibrations. Finally, a colored operad ${{\cal O}}$ is called [*cofibrant*]{} if the unique map $\ast \to  {{\cal O}}$ from the initial object $\ast$ is a cofibration.

Unfortunately, to the best of our knowledge, it is not yet established whether the category of all colored operads with the above (co)fibrations is a closed model category. For this reason, in our paper, we use the above terminology but not the full power of closed model categories.

We use the reversed grading on each homological complex. In particular, the homology groups $H_{{{\bullet}}}(X)$ of a topological space $X$ are concentrated in non-positive degrees and the Poincaré duality takes this unusual form: $$\label{PD}
H_{{{\bullet}}}(X) \cong 