Subsection 5.3. (0N67) · ScalingStacks

0N5L

Proposition 5.13. For $\mathfrak{g}$ a Lie algebra over $k$ , there is a natural equivalence of chain complexes over $k$ ,

\int_{M}\mathsf{C}_{\ast}^{\Lie}\bigl(\Mapc(\mathbb{R}^{n},\mathfrak{g})\bigr)~\simeq~\mathsf{C}^{\Lie}_{\ast}\bigl(\Mapc(M,\mathfrak{g})\bigr)~,

between the factorization homology of the Lie algebra chains of $\Omega^{n}{\mathfrak{g}}$ and the Lie algebra chains of the Lie algebra $\mathfrak{g}^{M^{+}}$ .

0N5M

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.

∎

0N5N

Remark 5.14. The $n$ -disk algebra $\mathsf{C}_{\ast}^{\Lie}\bigl(\Mapc(\mathbb{R}^{n},\mathfrak{g})\bigr)$ has an interesting separate interpretation that we state here, and prove as a separate work. There is a forgetful functor from $\mathcal{E}_{n}$ -algebras in chain complexes over $k$ to Lie algebras over $k$ (see [Coh] for an account at the level of homology). The adjoint functor theorem (Corollary 5.5.2.9 of [Lu1]) applies to this functor, and so there is an adjunction

\mathsf{U}_{n}\colon\Alg_{\Lie}({\sf Mod}_{k})~\rightleftarrows~\Alg_{\disk_{n}^{\sf fr}}({\sf Mod}_{k})\colon{\sf fgt}~.

In the case $n=1$ , this left adjoint $\mathsf{U}_{1}$ agrees with the familiar universal enveloping algebra functor. In general, there is an identification of $\disk_{n}^{\sf fr}$ -algebras,

\mathsf{U}_{n}\mathfrak{g}~\simeq~\mathsf{C}_{\ast}^{\Lie}\bigl(\Mapc(\mathbb{R}^{n},\mathfrak{g})\bigr)~,

through which Proposition 5.13 can be reformulated as an equivalence of chain complexes over $k$ :

\int_{M}\mathsf{U}_{n}\mathfrak{g}~\simeq~\mathsf{C}^{\Lie}_{\ast}\bigl(\Mapc(M,\mathfrak{g})\bigr)~.

5.3. Factorization homology from Lie algebras