ScalingStacks

Proof. Lie algebra chains defines a functor between $\infty$-categories $\mathsf{C}^{\Lie}_{\ast}:\Alg_{\Lie}({\sf Mod}_{k})\rightarrow{\sf Mod}_{k}$. This functor carries finite products of Lie algebras to finite tensor products of $k$-modules, which is to say that it is symmetric monoidal. Furthermore, this functor preserves geometric realizations, so Lemma 3.25 applies to give a natural identification: $\int_{M}\mathsf{C}_{\ast}^{\Lie}\bigl(\Mapc(\mathbb{R}^{n},\mathfrak{g})\bigr)\simeq\mathsf{C}_{\ast}^{\Lie}\bigl(\int_{M}\Mapc(\mathbb{R}^{n},\mathfrak{g})\bigr)$. The equivalence $\int_{M}\Mapc(\mathbb{R}^{n},\mathfrak{g})\simeq\Mapc(M,\mathfrak{g})$ now follows from the argument of nonabelian Poincaré duality (Corollary 4.6), only here we apply it in the usual abelian setting.⁶⁶ 6 See [AF2] for a complete account, where nonabelian Poincaré duality is an instance of a version of Poincaré/Koszul duality for Cartesian-presentable $\infty$-categories.

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