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

Π vert(P) A Π(P) ϵA Π(X) ΣA \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

T vertP 𝔞 flatformonfibers TP ϵ𝔞 formontotalspace TX Σ𝔞 characteristicforms \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

Ω vert (P) CE(𝔞) Ω (P) (A,F A) W(𝔞) Ω (X) inv(𝔞) \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 PP 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)