Boardman-Vogt construction for PROP(erads)

Question

Let $\left\lbrace \mathsf{O}(n)\right\rbrace_{n\in \mathbb{N}} $ be an operad in a symmetric monoidal category $(\mathsf{C},\otimes, \mathbf{1})$ which in addition has the structure of a model category (I think of topological spaces or chain complexes). In this case I think there exists the Boardman-Vogt construction (or $W$-construction) producing a new operad $\left\lbrace W\mathsf{O}(n)\right\rbrace_{n\in \mathbb{N}}$, which in modern language is a cofibrant replacement of $\mathsf{O}$.

Q: Basically my question is, if there exists a similar construction for properads or PROPs?

For $\mathsf{C}=\mathsf{Top}$ topological spaces the Boardman-Vogt construction can be carried out by introducing metric trees where internal edges of reduced trees carry a lenght map with codomain $[0,1]$. Hence the space of metrics $\mathrm{Met}(T)$ for a reduced tree with $n$ internal edges is the $n$-cube $I^n$. The operad $W\mathsf{O}$ then has arity $k$-space \begin{equation} W\mathsf{O}(k)=\bigsqcup_{T} \mathrm{Met}(T)\times \mathsf{O}(T)/ \sim \end{equation} where the disjoint union runs over isomorphism classes of reduced trees with $k$-leaves and $\mathsf{O}(T)=\bigsqcup_{v\in \mathrm{Vert}(T)}\mathsf{O}(in(v))$. The equivalence relation contracts edges of lenght 0 in metric trees.

So for properads is there something similar possible by replacing reduced trees with $k$-leaves with reduced planar graphs with $n$ incoming leaves and $m$ outgoing leaves?

I guess if the answer is yes algebras over the Boardman-Vogt construction of a properad are homotopy versions of the original properad algebras.

Answer 1

Such a construction is described in the preprint Boardman-Vogt resolutions of generalized Props in here. It uses the language of generalized props in the book A Foundation for PROPs, Algebras, and Modules. Operads, properads, props, and their colored variants are examples of generalized props. The Boardman-Vogt construction is defined in Chapter 3 of that preprint using a coend indexed by a category called the substitution category. Its categorical and homotopical properties are in Chapters 4-9. Theorem 7.3.2 says that, under suitable conditions, the Boardman-Vogt resolution is a cofibrant resolution.