For any smal category $A$, I shall write $\widehat A$ for the category $[A^{\text op}, \mathbf{Set}]$ of presheaves on $A$, and $y_A\colon A \to \widehat A$ for the Yoneda embedding relative to $A$.

I denote by $\Delta$ the category of simplices, by $\Omega$ the category of symmetric dendroids (or symmetric finite rooted trees) and by $\mathbf{Oper}$ the category of symmetric operads. The last two categories were introduced by I. Moerdijk and I. Weiss in WeissPhD and MW07; in the same articles they introduced the notion of Boardman-Vogt tensor product of operads $\otimes_{\text BV}\colon\mathbf{Oper}\times\mathbf{Oper}\to\mathbf{Oper}$, which endows the category of small operads with a symmetric closed monoidal structure.

As $\mathbf{Oper}$ is a Bénabou cosmos with the Boardman-Vogt tensor product, we can define a pair of adjoint functors $\tau_d\dashv N_d$ with the nerve-realization paradigm (using the fact the $\mathbf{Oper}$ is cocomplete) and a tensor product of dendroidal sets (i.e. presheaves on $\Omega$) $\otimes \colon\widehat \Omega\times \widehat\Omega\to \widehat\Omega$ defined as the left Kan extension of $N_d(-\otimes -)\colon \Omega\times \Omega\to \widehat\Omega$ along $y_\Omega$.

In §3.5 of the article HHM15 it is stated that the category of dendroidal sets $\widehat \Omega$ is weakly enriched over the cartesian closed category of simplical sets $\widehat \Delta$.

Thanks to the nerve-realization paradigm $\tau_d\dashv N_d$ (for the dendroidal-operadic case), it is enough to prove that there exists a natural isomorphism

$$ \varphi \colon y_\Delta\!\cdot\otimes(y_\Delta\!\cdot\otimes\, y_\Omega\cdot) \to N_d\bigl(\,\cdot\otimes_{\text BV}(\,\cdot\otimes_{\text BV}\cdot\,)\bigr):\Delta\times (\Delta\times \Omega) \to \widehat\Omega $$ and hence the weak enrichement follows by astract non-sense and the associativity of the Boardman-Vogt tensor product.

More generally, fixed an object $R$ of $\Omega$, the functor $\Omega^R\otimes\cdot\,\colon \widehat\Omega \to \widehat\Omega$ can be express as the left Kan extension of $N_d\circ R\otimes_{\text BV}\cdot$ along $y_\Omega$. Thus

$$ \bigl( \Delta^m\otimes (\Delta^n \otimes \Omega^T)\bigr)_S \cong \int^R \widehat \Omega(\Omega^R, \Delta^n \otimes \Omega^T)\times N_d([m]\otimes_{\text BV} R)_S\\ \phantom{AALA\bigl( \Omega^R\otimes (\Omega^S \otimes \Omega^T)\bigr)_Q}\cong \int^R \mathbf{Oper}(R, [n]\otimes_{\text BV} T) \times \mathbf{Oper}(S, [m]\otimes_{\text BV} R)\\ $$ for any $[m], [n]$ in $\Delta$, $T$ in $\text{Ob}\Omega$, and where we have denoted by $\Omega^T$ the presheaf $y_\Omega T$ on $\Omega$. Amazingly enough,

$$ \int^{R\in \Omega} \mathbf{Oper}(R, [n]\otimes_{\text BV} T) \times \mathbf{Oper}(S, [m]\otimes_{\text BV} R) \cong \int^{\mathcal P \in \mathbf{Oper}} \mathbf{Oper}(\mathcal P, [n]\otimes_{\text BV} T) \times \mathbf{Oper}(S, [m]\otimes_{\text BV} \mathcal P) $$ where the last term is exaclty the coend form (Yoneda expansion) of the functor $N_d([m]\otimes_{\text BV}\cdot\,)_S\colon \mathbf{Oper} \to \mathbf{Set}$ evaluated in $[n]\otimes_{\text BV} T$.

Using the adjunction $\tau_d\dashv N_d$ and the fact that $\tau_d$ distributes over the tensor product of dendroidal sets, it is possible to get then a natural transformation $$ \alpha\colon (y_\Delta\cdot\times y_\Delta\cdot\,)\otimes y_\Omega\cdot \to y_\Delta\cdot\otimes (y_\Delta\cdot\otimes y_\Omega\cdot\,) : \Delta \times \Delta \times \Omega \to \widehat\Omega\ . $$

Question 1. What is an example of a pair of operads $\mathcal{P, Q}$ in $\text{Ob}\mathbf{Oper}$ such that $N_d(\mathcal P\otimes_{\text BV}\mathcal Q)$ is not isomorphic to $N_d(\mathcal P) \otimes N_d(\mathcal Q)$, i.e. for which the nerve does not distribute over the Boardman-Vogt tensor product?

It is stated in §3.5 of HHM15 that the category $\widehat\Omega$ of dendroidal sets is not a monoidal category with the tensor product, as the associativity fails. Indeed, they prove a weak associativity for representable presheaves, as I worked out above, and then it is stated that extending by colimits to all triple of presheaves on $\Omega$ that particular natural isomorphism between triple of representable presheaves, then the comparison morphism is not an isomorphism in general.

Question 2. What is an example of a triple of presheaves $X, Y, Z$ on $\Omega$ such that the comparison morphism built in HHM15 is not an isomorphism? An example for which $X, Y$ are actually presheaves on $\Delta$ would be appreciated even more.

Question 3. Why cannot exist any other associator $\alpha$ (natural isomorphism for the associativity) making $(\widehat \Omega, \otimes, \alpha, \lambda, \rho, \underline{hom}_\Omega)$ a closed symmetric monoidal category?

For any presheaves $X, Y$ on $\Omega$ and any object $T$ of $\Omega$, we define $\underline{hom}_\Omega(X, Y)_T = \widehat\Omega(\Omega^T\otimes X, Y)$.

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    $\begingroup$ I assume you wrote the right coend. Then the ninja Yoneda lemma tells you that $$ (\Delta^m\otimes (\Delta^n\otimes \Omega^T))_R \cong \mathbf{Opd}(R, [m]\otimes ([n]\otimes T))$$ which is, I think, something less scary. $\endgroup$ – Fosco Loregian Feb 19 '15 at 22:47
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    $\begingroup$ I think it all boils down in defining us the functor $N_d$: if it sends $\mathcal C$ to the presheaf $R\mapsto \mathbf{Opd}(R, \mathcal C)$, I think we are done! $\endgroup$ – Fosco Loregian Feb 19 '15 at 22:57
  • $\begingroup$ Right! I improved the questions based on this remark. $\endgroup$ – Andrea Gagna Feb 20 '15 at 15:57
  • $\begingroup$ I copied the same thread on MO, as well. $\endgroup$ – Andrea Gagna Feb 20 '15 at 17:03
  • $\begingroup$ A complete and nice answer by G. Heuts can be found here. $\endgroup$ – Andrea Gagna Feb 25 '15 at 14:08

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