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Let $k$ be a field of characteristic 0. Let $\mathtt{DGA}_{k}$ denote the category of DG algebras and $\mathtt{DGLA}_{k}$ denote the category of DG Lie algebras. It is well known that there are model structures on them that the weak equivalences are the quasi-isomorphisms and the fibrations are the maps which are degreewise surjective.

We have a pair of adjoint functors (in fact a Quillen pair) $$\mathcal{U}: \mathtt{DGLA}_{k} \leftrightarrows\mathtt{DGA}_{k} : \mathcal{Lie}$$ Where $\mathcal{U}$ is the universal enveloping algebra functor and $\mathcal{Lie}$ is the forgetful functor that sends every DG algebra to the DG Lie algebra defined by commutator bracket.

I am thinking about some derived functors on those model categories. One thing I need to convince myself is that $\mathcal{U}$ preserves quasi-isomorphism. The only proof I can find is in Félix, Halperin and Thomas' book Rational Homotopy Theory, but it is kind of brief and I can't fully understand the argument.

Because we are working over a field of characteristic 0, the homology $H$ will be a direct summand of our DG Lie algebra $L$ as complexes. And by PBW theorem, we can identify $\mathcal{U}(L)$ with $\mathrm{Sym}(L)$ as complexes. But now we need to show that $\mathrm{Sym}$ preserves quasi-isomorphism, and I don't know how to show it. Can anyone give me some hints or some good references? Thank you very much.

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    $\begingroup$ For chain complexes over a field $k$: $C \otimes_k - $ preserves quasi-isomorphisms. Use this to show tensor powers preserve quasi-isomorphisms. Then in characteristic zero, taking $S_n$ invariants is exact, so we get that $Sym^n$ preserves quasi-isomorphisms. $\endgroup$ May 18, 2016 at 20:58
  • $\begingroup$ The adjunction $\mathrm{Hom}_{\mathtt{DGCA}_{k}}(\mathrm{Sym}(V), A)\cong \mathrm{Hom}_{\mathtt{GrVect}_{k}}(V, A)$ implies that $\mathrm{Sym}$ is right exact. To show it is also left exact, up to isomorphism, we just need to show $\mathrm{Sym}(W)\subset \mathrm{Sym}(V)$ provided that $W\subset V$. It is straightforward if we view $\mathrm{Sym}(V)$ (resp., $\mathrm{Sym}(W)$) as the polynomial algebra generated by the basis of $V$ (resp., $W$) . However, is the argument also correct in the case of characteristic $p$? $\endgroup$ May 19, 2016 at 1:34
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    $\begingroup$ It seems like my argument above is totally wrong. Since $\mathtt{DGCA_{k}}$ is not an abelian category, the notion of right or left exact are not described via short exact sequence. $\endgroup$ May 19, 2016 at 20:05

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