Symmetric monoidal structure on algebras

Question

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

Answer 1

This is worked out in Higher Algebra, example 3.2.4.4.

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 $\{A_i(o_j)\}_{(i,j)\in I\times J}$.