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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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    $\begingroup$ A similar thread (that eventually has become exactly the same) is also on MSE. $\endgroup$ – Andrea Gagna Feb 20 '15 at 17:02
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Associativity of the tensor product in fact does not hold even for representable presheaves, which addresses your Questions 2 and 3. Proposition 3.6.9 and its proof (in HHM, as you cite) give a typical counterexample, worked out explicitly there.

As to Question 1: consider the associative operad $\mathbf{A}$. Tensoring that with itself gives the commutative operad (essentially by the Eckman-Hilton argument), but the tensor product $N_d(\mathbf{A}) \otimes N_d(\mathbf{A})$ is actually a model for the homotopy-coherent nerve of the $\mathbf{E}_2$-operad.

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  • $\begingroup$ Excellent, many thanks! I hadn't reached §3.6 of your nice paper yet. Now I also see my miscalculations with coend: I will edit my question. I see that your counter-example to Question 2 and 3 also gives one for Question 1: namely, $(\Delta^1\times \Delta^1) \otimes C_2$ is $N_d([1]\times [1]) \otimes N_d\bigl(\Omega(C_2)\bigr)$ which is not the same as $N_d\bigl(([1]\times [1])\otimes_{\text BV} \Omega(C_2)\bigr) \cong \Delta^1 \otimes (\Delta^1 \otimes C_2)$ (it is actually a proper subpresheaf of the latter one). $\endgroup$ – Andrea Gagna Feb 25 '15 at 11:39
  • $\begingroup$ I really like your answer to Q1, though, since I have been wondering for a while what is the tensor product of $P=N_d(\mathbf A)$ (which is the presheaf of planar structures: $\mathbf{dSet}/P \simeq \mathbf{pdSet}$) with itself. $\endgroup$ – Andrea Gagna Feb 25 '15 at 11:49

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