Schreiber Cartan-Ehresmann ∞-connection

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 $Lie : sSh(C) \to sSh(C)$ that sends the Ehresmann ∞-connection diagram of ∞-Lie groupoids

$$\array{ \Pi_{vert}(P) &\stackrel{}{\to}& A \\ \downarrow && \downarrow \\ \Pi(P) &\stackrel{\nabla}{\to}& \epsilon A \\ \downarrow && \downarrow \\ \Pi(X) &\to& \Sigma A }$$

to the diagram

$$\array{ T_{vert} P &\stackrel{}{\to}& \mathfrak{a} && flat form on fibers \\ \downarrow && \downarrow \\ T P &\stackrel{\nabla}{\to}& \epsilon \mathfrak{a} && form on total space \\ \downarrow && \downarrow \\ T X &\to& \Sigma \mathfrak{a} && characteristic forms }$$

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

The corresponding diagram of Chevalley-Eilenberg algebras is

$$\array{ \Omega^\bullet_{vert}(P) &\stackrel{}{\leftarrow}& CE(\mathfrak{a}) \\ \uparrow && \uparrow \\ \Omega^\bullet(P) &\stackrel{(A,F_A)}{\leftarrow}& \mathrm{W}(\mathfrak{a}) \\ \uparrow && \uparrow \\ \Omega^\bullet(X) &\leftarrow& inv(\mathfrak{a}) }$$

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.