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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"?

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  • $\begingroup$ 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 $\endgroup$
    – Bertram Arnold
    Commented Dec 17, 2020 at 10:24
  • $\begingroup$ 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". $\endgroup$
    – Minkowski
    Commented Dec 17, 2020 at 11:15
  • $\begingroup$ 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. $\endgroup$
    – Bertram Arnold
    Commented Dec 17, 2020 at 11:23

1 Answer 1

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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.

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