nLab Weil algebra abstractly -- table

Chevalley-Eilenberg algebra CE $\leftarrow$ Weil algebra W $\leftarrow$ invariant polynomials inv

differential forms on moduli stack $\mathbf{B}G_{conn}$ of principal connections (Freed-Hopkins 13):

$$\array{ CE(\mathfrak{g}) &\simeq& \Omega^\bullet_{li \atop cl}(G) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\simeq & \Omega^\bullet(\mathbf{E}G_{conn}) & \simeq & \Omega^\bullet(\mathbf{\Omega}(-,\mathfrak{g})) \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\simeq& \Omega^\bullet(\mathbf{B}G_{conn}) & \simeq & \Omega^\bullet(\mathbf{\Omega}(-,\mathfrak{g})/G) }$$