Schreiber
Cartan-Ehresmann ∞-connection

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

Idea

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

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

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

$$\begin{array}{ccc}{\Omega }_{\mathrm{vert}}^{•}(P)& \stackrel{}{\leftarrow }& \mathrm{CE}(𝔞)\\ \uparrow & & \uparrow \\ {\Omega }^{•}(P)& \stackrel{(A,F_A)}{\leftarrow }& \mathrm{W}(𝔞)\\ \uparrow & & \uparrow \\ {\Omega }^{•}(X)& \leftarrow & \mathrm{inv}(𝔞)\end{array}$$\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 $\mathrm{\infty }$-connection . It appears in this form in SaScStI.