In "The Boardman-Vogt resolution of operads in monoidal model categories," the authors construct factorizations of sufficiently nice operad maps $P\to Q$ into a cofibration followed by a weak equivalence $P\rightarrowtail W(H,Q)_P\stackrel{\sim}{\to}Q$. This construction depends on a choice of interval $H$. When $P$ is the unit operad $I$, this is called $W(H,Q)$ and is a cofibrant replacement for $Q$. My question is about the relative construction when $P\ne I$.
The factorization $W(H,Q)\_P$ is constructed as a sequential colimit of pushouts. For simplicity, concentrate everything in arity $1$ so that we can ignore $\Sigma_n$ actions; then these pushouts are of the form
$H(T)\otimes\underline{Q}(T)\gets {(H\otimes Q)}^-\_P(T)\to W_{k-1}(H,Q)_P.$
The pushout of this diagram (taken as $T$ ranges over all appropriate trees) is $W_{k}(H,Q)_P$. It is my understanding that the middle entry of this pushout is the colimit over a cube with the final corner omitted, and that the left entry is that final corner. The map to the right factor, which is constructed inductively,
informally puts edge-lengths to 0, whenever the vertices of the edge are both labelled by elements of P, and then applies the corresponding attaching map of the absolute W-construction.
The middle entry of the pushout is somewhat involved to describe explicitly; it occurs on p833--834 and relies on p819--821 in the published version of the paper. Let's pick an example,and I'll do a calculation, and hopefully someone can point out the mistake I'm making.
Let's look at the bivalent tree with four vertices $(v_1,v_2,v_3,v_4)$ and three internal edges $(e_{12}, e_{23}, e_{34})$. I believe that one part of the diagram making up the middle entry is:
$I_{12}\otimes H_{23}\otimes I_{34}\otimes Q_1\otimes I_2\otimes I_3\otimes Q_4$
and that this maps along the edges of the cube into both
$I_{12}\otimes H_{23}\otimes I_{34}\otimes Q_1\otimes I_2\otimes Q_3\otimes Q_4$
and
$I_{12}\otimes H_{23}\otimes I_{34}\otimes Q_1\otimes P_2\otimes P_3\otimes Q_4$
In the first case, the attaching map to $W_{k-1}(H,Q)\_P$ collapses the edge $e_{34}$, getting the vertex label $Q_3\otimes Q_4$ since the edge between them is labeled by $I$ and also forgets $v_2$ because it is labeled by $I$, using the product on $H$ to get the label on the resulting edge. So we get a tree with vertices $(v_1, v_{34})$ and edges $(e_{134})$. These are labeled by $Q$, $Q$, and $H$.
In the second case, the attaching map first collapses the edge $e_{23}$ because it is between two elements of $P$, composing $P_2\otimes P_3\to P$, and then collapses the other two edges because they are of length $0$, getting $Q_1\otimes P\otimes Q_4\to Q$. In this case, we get a tree with only one vertex and no internal edges.
In order for the attaching map to be well-defined on the colimit defining the middle entry, if we begin in $I_{12}\otimes H_{23}\otimes I_{34}\otimes Q_1\otimes I_2\otimes I_3\otimes Q_4$ then we must get the same eventual answer in $W(H,Q)_P$ whichever one of these procedures we choose. Then, shifting for concreteness' sake to, say, the topological category and the Boardman-Vogt interval, we get that the formal composition on the two vertex tree $q_1, q_2$ with edge length $r$ is equivalent to the composition $q_1\circ q_2$ on the one-vertex tree. This is a problem because it means that the entire structure collapses to $Q$, which is not in general sufficiently cofibrant.
