# Symmetric monoidal structure on algebras

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

Concretely, $$\mathrm{Alg}_{\mathcal{O}}(\mathcal{C})^\otimes$$ is defined as follows: it is the simplicial set over $$\mathrm{Fin}_\ast$$ such that for any simplicial set $$K\to \mathrm{Fin}_\ast$$, the set of maps $$K\to \mathrm{Alg}_{\mathcal{O}}(\mathcal{C})^\otimes$$ over $$\mathrm{Fin}_\ast$$ is equivalent to the set of diagrams $$\require{AMScd} \begin{CD} K\times\mathcal{O}^\otimes @>>> \mathcal{C}^\otimes\\ @VVV @VVV \\ \mathrm{Fin}_\ast\times\mathrm{Fin}_\ast @>{\wedge}>> \mathrm{Fin}_\ast \end{CD}$$ such that for every $$k\in K$$ restriction of the top arrow to $$\{k\}\times \mathcal{O}^\otimes$$ sends inert morphisms to inert morphisms. Here the bottom arrow is the smash product of pointed sets, sending $$(I_+,J_+)$$ to $$(I\times J)_+$$.
If you consider the fiber over $$1_+$$, you see that it is exactly $$\mathrm{Alg}_\mathcal{O}(\mathcal{C})$$, moreover for every $$o\in\mathcal{O}$$ there is a canonical symmetric monoidal functor $$\mathrm{Alg}_\mathcal{O}(\mathcal{C})^\otimes\to \mathcal{C}^\otimes$$, given by taking the fiber over $$o$$ of the previous diagram, thus showing that this deserves the name "pointwise tensor product".
This might seem an abstract way of defining it, and it is, but if you reflect a bit on what it is doing you'll see that it makes sense: the smash product of pointed sets is precisely encoding the multiplication you want on the algebra $$A\otimes B$$. For example, for an associative algebra, the product on $$A\otimes B$$ is given by $$(A\otimes B)\otimes (A\otimes B)\cong (A\otimes A)\otimes (B\otimes B)\to A\otimes B$$ as desired.
Another way of thinking about this is that $$\mathrm{Alg}_\mathcal{O}(\mathcal{C})^\otimes$$ is the universal object $$\mathcal{A}^\otimes$$ together with an "evaluation map" $$\mathcal{A}^\otimes\times\mathcal{O}^\otimes\to C^\otimes$$ sending $$(\{A_i\}_{i\in I},\{o_j\}_{j\in J})$$ to $$\{A_i(o_j)\}_{(i,j)\in I\times J}$$.