ScalingStacks

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 $k$:

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