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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 10th 2015
   • (edited Nov 10th 2015)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have (finally) added some pointers to the result of Freed-Hopkins 13 to relevant $n$Lab entries. Mostly at _[Weil algebra -- characterization in the smooth infinity-topos](http://ncatlab.org/nlab/show/Weil+algebra#CharacterizationInSmoothTopos)_ also at _[invariant polynomials -- As differential forms on the moduli stack of connections](http://ncatlab.org/nlab/show/invariant+polynomial#AsFormsOnBGconn)_ pointing out that this adds further rationalization to the construction of [connections on principal infinity-bundles -- via Lie integration](http://ncatlab.org/nlab/show/connection+on+a+smooth+principal+infinity-bundle#ByLieIntegration). In making these edits, I have created and then used a little table-for-inclusion * _[[Weil algebra abstractly -- table]]_ Presently this displays as follows: *** **[[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](Weil+algebra#FreedHopkins13)): $$ \array{ CE(\mathfrak{g}) &\simeq& \Omega^\bullet_{li \atop cl}(G) \\ \uparrow && \uparrow \\ W(\mathfrak{g}) &\simeq & \Omega^\bullet(\mathbf{E}G_{conn}) & \simeq & \Omega^\bullet(\mathbf{\Omega}(-,\mathfrak{g})) \\ \uparrow && \uparrow \\ inv(\mathfrak{g}) &\simeq& \Omega^\bullet(\mathbf{B}G_{conn}) & \simeq & \Omega^\bullet(\mathbf{\Omega}(-,\mathfrak{g})/G) } $$

   I have (finally) added some pointers to the result of Freed-Hopkins 13 to relevant $n$Lab 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

   • Weil algebra abstractly – table

   Presently this displays as follows:

   ---

   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):

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