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, 



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> 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.
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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.



How does Berger-Moerdijk's relative Boardman-Vogt work?
It turns out that the construction as written in the paper is incorrect, but there is a fix known to the authors. Instead of collapsing $P\otimes H\otimes P$ to $P$, one only collapses $I\otimes P\otimes H\otimes P\otimes I$ (where the $I$ factors are included in $H$ with length 1, not 0). This gets rid of the problem above and the rest of the construction checks out. Many thanks to Gijs Heuts for communicating this to me.


