Recall that an *operad* (in vector spaces, say) $P$ consists of a collection of vector spaces $P(n)$ for $n\geq 0$, such that $P(n)$ is equipped with an action by the symmetric group $S\_n$, with maps $P(n) \otimes P(k\_1) \otimes \dots \otimes P(k\_n) \to P(k\_1+\dots k\_n)$ for any $n,k\_1,\dots,k\_n$, subject to natural associativity constraints. Recall also that a *$P$-algebra* for an operad $P$ consists of a vector space $V$ and for each $n$ a map $P(n) \to \hom(V^{\otimes n},V)$ subject to natural associativity constraints. Recall finally that for any operad $P$, the functor from vector spaces to $P$-algebras that is adjoint to the Forgetful functor is:
$$ \operatorname{Free}: V \mapsto \bigoplus\_n \bigl(P(n) \otimes V^n \bigr)\_{S\_n} $$
where by $W\_{S\_n}$ I mean the coinvariants of the $S\_n$-module $W$, i.e. $W\_{S\_n} = W \otimes\_{\mathbb{K} S\_n} 1$.


Let me switch to working with chain complexes rather than vector spaces. Then I can do a more refined operation than taking coinvariants. Namely, I can use the *derived* functor of coinvariants: you replace $W$ by a projective resolution, tensor, and consider the result up to quasiisomorphism. Which is to say "the complex that computes $\operatorname{Tor}$".



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> **Question:** What is the meaning of the ($\infty$-)functor that takes derived coinvariants rather than coinvariants in the above formula? What relationship does it play to the operad, etc.?
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The type of answer I'm looking might be the following: "The $(\infty,1)$-functor so described is adjoint to the functor that forgets from the $(\infty,1)$-category of strongly homotopy $P$-algebras to the $(\infty,1)$-category of chain complexes."


Another direction that you could take the following question might be:



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> **Question 2:** What type of "Koszul duality" statements are there using the "derived coinvariants" version of $\operatorname{Free}$?
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Is there a "derived" Free $P$-algebra functor for an operad $P$?
1. Yes this is an expression of the free algebra functor, left adjoint to the forgetful functor, see Section 3.1.3 of Lurie's Higher Algebra.
2. Koszul duality in the (oo,1)-setting is treated very nicely in the work of John Francis, see in particular math/1104.0181, where eg the relation with deformation theory is explained (the tangent complex to an augmented O-algebra carrying an action of the Koszul dual operad, etc)


