I'm reading this paper by Hinich and he uses two notations involving the symmetric group $S_n$ that he doesn't clarify, so I assume that they are standard, but I don't know what they mean.
The first of them appears at the end of page 3:
Let $\mathscr{O}$ be an operad in a tensor category $\mathscr{A}$. Let $V$ be an $\mathbb{S}$-object in $\mathscr{A}$. The free $\mathscr{O}$-algebra generated by $V$ is defined to be $$ \Bbb{F}_{\mathscr{O}}(V)=\bigoplus_{n\geq 0}\mathscr{O}(n)\otimes_{S_n}V^{\otimes n} $$ with a canonical $\mathscr{O}$-algebra structure.
I think it doesn't make sense to take tensor product with respect to $S_n$ (as in $R$-modules with respect to $R$) so I guess it has to do with the $S_n$-equivariance of the action of the operad $\mathscr{O}$ on the algebra, but I'm not sure.
Then there is a dual definition
If $V$ is an $\mathbb{S}$-object in $\mathscr{A}$, the cofree coalgebra cogenerated by $V$ is defined to be $$ \Bbb{F}^*_{\mathscr{C}}(V)=\bigoplus_{n\geq 0}\left(\mathscr{C}(n)\otimes V^{\otimes n}\right)^{S_n} $$
In this case I have no clue what it means. It cannot be the maps to $S_n$ since for $n=1$ it is used a few pages later (after equation (9)) that it is just $V$. Maybe it is just a direct sum of $n!$ copies, but that's just a guess.
I hope someone recognizes these notations. Thank you.
I looked it up in Algebraic Operads by Loday and Vallette (chapter 5).
The first one is the tensor product taken under the action of $S_n$. On $\mathscr{O}(n)$ it is the action that has by definition of $\Bbb{S}$-object and in $V^{\otimes n}$ is a rearrangement of the coordinates.
The second one is the space of invariant elements under this action.