If $\mathbf{P}$ is a (coloured) operad, one can build a topological operad $W(\mathbf{P})$ called the $W$-construction or the Boardman-Vogt resolution of $\mathbf{P}$. Let me denote the resulting map of operads $\varepsilon: W(\mathbf{P}) \to \mathbf{P} $.


My question is: if $\mathbf{P} = \mathbf{E}\_2$ the little 2-cubes (or 2-discs) operad, how does $W(\mathbf{E}\_2)$ look like? And even more of my interest: if $A$ is a $\mathbf{P}$-algebra, how does the $W(\mathbf{E}\_2)$-algebra $\varepsilon^\* (A)$ look like?


**Edit:** More concretely, how should $\varepsilon^\* (A)$ be understood as an $\mathbf{E}\_2$-algebra "up to homotopy"?



Boardman-Vogt resolution of the little 2-cubes operad
In the more general setting of a symmetric monoidal category $\mathsf{M}$ and a general colored operad $\mathsf{O}$, the structure of an $\mathsf{O}$-algebra $X$ regarded as a $\mathsf{WO}$-algebra is described explicitly in the book [Homotopical Quantum Field Theory](https://www.worldscientific.com/worldscibooks/10.1142/11626) (called HQFT below), Corollary 7.2.9. The arxiv version is [here](https://arxiv.org/abs/1802.08101). When $\mathsf{M}$ is a suitable category of topological spaces $\mathsf{Top}$, this pullback algebra structure is as Bertram Arnold described in the comments above, where you first forget about the edge lengths. In the general context of $\mathsf{M}$, the interval $[0,1]$ is replaced by a commutative segment $\mathsf{J}$ (Definition 6.2.1 in HQFT), which provides a notion of homotopy and comes with a counit map to the monoidal unit of $\mathsf{M}$. In $\mathsf{Top}$, this counit is $[0,1] \to \*$.


To understand Corollary 7.2.9, you will first need to read Theorem 7.2.1 in HQFT, which explicitly describes the structure of a $\mathsf{WO}$-algebra using the formalism of trees and commutative segment. That theorem is one way to formally say that a $\mathsf{WO}$-algebra is an $\mathsf{O}$-algebra up to homotopy. In HQFT Definition 6.3.1, the Boardman-Vogt construction $\mathsf{WO}$ is defined as an entrywise coend indexed by a parameter category built from trees. You can apply that definition and the results mentioned above to the little 2-cube operad or any other operad. A number of examples can be found in HQFT.


