# Schreiber Cartan-Ehresmann ∞-connection

differential cohomology in an (∞,1)-topos -- survey

structures in an (∞,1)-topos

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∞-Lie theory

## symplectic ∞-geometry

A Cartan-Ehresmann $\infty$-connection is the the expression of an Ehresmann ∞-connection in terms of ∞-Lie algebroid valued differential forms.

# Idea

A Cartan -Ehresmann $\infty$-connection is what is obtained from an Ehresmann ∞-connection after applying the ∞-Lie differentiation functor $\mathrm{Lie}:\mathrm{sSh}\left(C\right)\to \mathrm{sSh}\left(C\right)$ that sends the Ehresmann ∞-connection diagram of ∞-Lie groupoids

$\begin{array}{ccc}{\Pi }_{\mathrm{vert}}\left(P\right)& \stackrel{}{\to }& A\\ ↓& & ↓\\ \Pi \left(P\right)& \stackrel{\nabla }{\to }& ϵA\\ ↓& & ↓\\ \Pi \left(X\right)& \to & \Sigma A\end{array}$\array{ \Pi_{vert}(P) &\stackrel{}{\to}& A \\ \downarrow && \downarrow \\ \Pi(P) &\stackrel{\nabla}{\to}& \epsilon A \\ \downarrow && \downarrow \\ \Pi(X) &\to& \Sigma A }

to the diagram

$\begin{array}{ccccc}{T}_{\mathrm{vert}}P& \stackrel{}{\to }& 𝔞& & \mathrm{flat}\mathrm{form}\mathrm{on}\mathrm{fibers}\\ ↓& & ↓\\ TP& \stackrel{\nabla }{\to }& ϵ𝔞& & \mathrm{form}\mathrm{on}\mathrm{total}\mathrm{space}\\ ↓& & ↓\\ TX& \to & \Sigma 𝔞& & \mathrm{characteristic}\mathrm{forms}\end{array}$\array{ T_{vert} P &\stackrel{}{\to}& \mathfrak{a} && flat form on fibers \\ \downarrow && \downarrow \\ T P &\stackrel{\nabla}{\to}& \epsilon \mathfrak{a} && form on total space \\ \downarrow && \downarrow \\ T X &\to& \Sigma \mathfrak{a} && characteristic forms }

of ∞-Lie algebroids that encodes a system of ∞-Lie algebroid valued differential forms.

The corresponding diagram of Chevalley-Eilenberg algebras is

$\begin{array}{ccc}{\Omega }_{\mathrm{vert}}^{•}\left(P\right)& \stackrel{}{←}& \mathrm{CE}\left(𝔞\right)\\ ↑& & ↑\\ {\Omega }^{•}\left(P\right)& \stackrel{\left(A,{F}_{A}\right)}{←}& \mathrm{W}\left(𝔞\right)\\ ↑& & ↑\\ {\Omega }^{•}\left(X\right)& ←& \mathrm{inv}\left(𝔞\right)\end{array}$\array{ \Omega^\bullet_{vert}(P) &\stackrel{}{\leftarrow}& CE(\mathfrak{a}) \\ \uparrow && \uparrow \\ \Omega^\bullet(P) &\stackrel{(A,F_A)}{\leftarrow}& \mathrm{W}(\mathfrak{a}) \\ \uparrow && \uparrow \\ \Omega^\bullet(X) &\leftarrow& inv(\mathfrak{a}) }

Such a diagram of ∞-Lie algebroid valued differential forms on a principal ∞-bundle $P$ we call a Cartan-Ehresmann $\infty$-connection . It appears in this form in SaScStI.

Revised on October 20, 2009 10:42:22 by Urs Schreiber (131.211.241.147)