I stuck at a relatively simple thing of formalization in infinity setting. I use here the formalism of quasi categories, i.e. simplicial sets with inner horn fillings.

Suppose $O^{\otimes}$ is an infinity operad and $C^{\otimes}$ a symmetric monoidal infinity category. Then I read that $Alg_O(C^{\otimes})$ can be endowed with a symmetric monoidal structure by "pointwise tensor product".

If everything was strict, it would be clear that one should put $(F \otimes G)(C) = F(C) \otimes G(C)$. With higher homotopies, I don't know how to move.

The first attempt I did is to look for a functor $Alg_O(C^{\otimes}) \to N(Fin_*)$ that encodes the pointwise thing, hoping that somehow the underlying category would be $Fun(O, C)$ (functors between the underlying categories). But if we think about the projection to $\to N(Fin_*)$ as the "degree" or the function "how many objects are there", then an algebra $O^{\otimes} \to C^{\otimes}$ does not have a precise degree: it simply preserves the degree of the involved objects.

The second attempt I made is to construct $Alg_O(C^{\otimes})^{\otimes}$ such that the underlying category is $Alg_O(C^{\otimes})$. In case $Alg_O(C^{\otimes})$ was strict, i know that the $\otimes$ construction is something like the set of tuples $(\langle n \rangle, x_1, \ldots, x_n)$ with morphisms given by $$ Hom ( (\langle n \rangle, x_1, \ldots, x_n), (\langle m \rangle, y_1, \ldots, y_m) ) = \coprod_{\alpha: \langle n \rangle \to \langle m \rangle } \prod_{i=1}^m Hom( \otimes_{\alpha(j)=i} x_j, y_i) $$

Also, in case $Alg_O(C^{\otimes})$ was a simplicial category, we could promote this definition to a simplicial categorical one by susbsituting hom-sets with map-ssets.

Nevertheless, in case the higher information is described as a simplicial set (as it is in my case), I don't know how to put it itno the picture. Any hints? Thanks!

Andrea

I stuck at a relatively simple thing of formalization in infinity setting. I use here the formalism of quasi categories, i.e. simplicial sets with inner horn fillings.

Suppose $O^{\otimes}$ is an infinity operad and $C^{\otimes}$ a symmetric monoidal infinity category. Then I read that $Alg_O(C^{\otimes})$ can be endowed with a symmetric monoidal structure by "pointwise tensor product".

If everything was strict, it would be clear that one should put $(F \otimes G)(C) = F(C) \otimes G(C)$. With higher homotopies, I don't know how to move.

The first attempt I did is to look for a functor $Alg_O(C^{\otimes}) \to N(Fin_*)$ that encodes the pointwise thing, hoping that somehow the underlying category would be $Fun(O, C)$ (functors between the underlying categories). But if we think about the projection to $\to N(Fin_*)$ as the "degree" or the function "how many objects are there", then an algebra $O^{\otimes} \to C^{\otimes}$ does not have a precise degree: it simply preserves the degree of the involved objects.

The second attempt I made is to construct $Alg_O(C^{\otimes})^{\otimes}$ such that the underlying category is $Alg_O(C^{\otimes})$. In case $Alg_O(C^{\otimes})$ was strict, i know that the $\otimes$ construction is something like the set of tuples $(\langle n \rangle, x_1, \ldots, x_n)$ with morphisms given by $$ Hom ( (\langle n \rangle, x_1, \ldots, x_n), (\langle m \rangle, y_1, \ldots, y_m) ) = \coprod_{\alpha: \langle n \rangle \to \langle m \rangle } \prod_{i=1}^m Hom( \otimes_{\alpha(j)=i} x_j, y_i) $$

Also, in case $Alg_O(C^{\otimes})$ was a simplicial category, we could promote this definition to a simplicial categorical one by substituting hom-sets with map-ssets.

Nevertheless, in case the higher information is described as a simplicial set (as it is in my case), I don't know how to put it into the picture. Any hints? Thanks!

Andrea