Boardman-Vogt resolution of the little 2-cubes operad

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"?

• It's not really clear what a non-formal answer to this question would be - for any operad $P$, $W(P)$ is given by trees whose vertices are labeled with operations of the operad and whose edges are labeled with lengths in $[0,1]$, subject to the relation which contracts an edge of length $0$ and composes the operations using the operad structure. The map down to $P$ is given by forgetting the lengths and composing along the tree. One interesting feature for $P = E_n$ is that $W(E_n)$ maps to the so-called Fulton-MacPherson operad, compare Proposition 3.9 in arxiv.org/abs/math/9907073 Dec 17, 2020 at 10:24
• Possibly an example of how an algebra over the little 2-cubes operad (essentially $\Omega^2 X$) becomes an algebra over the $W(E_2)$ operad would clarify things a lot. What I would like to understand is how the new algebra should be regarded as a $E_2$-algebra "up to homotopy". Dec 17, 2020 at 11:15
• Given a $P$-algebra $A$, it acquires a $W(P)$-algebra structure via $W(P)(n)\times A^n\to P(n)\times A^n\to A$, where the first map is the operad morphism $W(P)\to P$ I described in the previous comment and the second is the $P$-algebra structure. You can think of a $W(P)$-algebra as a "$P$-algebra up to coherent homotopy"; in particular, this example shows that every strict $P$-algebra canonically defines such a thing. It is the other way around, i.e. turning a $W(P)$-algebra into a $P$-algebra, which is in general impossible. Dec 17, 2020 at 11:23

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 (called HQFT below), Corollary 7.2.9. The arxiv version is here. 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.