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I have (finally) added some pointers to the result of Freed-Hopkins 13 to relevant nLab entries.
Mostly at Weil algebra – characterization in the smooth infinity-topos
also at invariant polynomials – As differential forms on the moduli stack of connections
pointing out that this adds further rationalization to the construction of connections on principal infinity-bundles – via Lie integration.
In making these edits, I have created and then used a little table-for-inclusion
Presently this displays as follows:
Chevalley-Eilenberg algebra CE ← Weil algebra W ← invariant polynomials inv
differential forms on moduli stack BGconn of principal connections (Freed-Hopkins 13):
CE(𝔤)≃Ω•licl(G)↑↑W(𝔤)≃Ω•(EGconn)≃Ω•(Ω(−,𝔤))↑↑inv(𝔤)≃Ω•(BGconn)≃Ω•(Ω(−,𝔤)/G)1 to 1 of 1