Connections on a principal Lie groupoid bundle and representations up to homotopy

Abstract

A Lie groupoid principal $\mbbX$ bundle is a surjective submersion $\ddot{I}€:P\to M$ with an action of $\mathbb{X}$ on $P$ with certain additional conditions. This paper offers a suitable definition for the notion of a connection on such bundles. Although every Lie groupoid $\mathbb{X}$ has its associated Lie algebroid $A:=1^{\ast }\ker ds\to X_0$, it does not admit a natural action on its Lie algebroid. There is no natural action of $\mathbb{X}$ on $TP$ either. Choosing a connection $\mathbb{H}\subset T{X}_1$ on the Lie groupoid $\mathbb{X},$ and considering its induced action up to homotopy of $\mathbb{X}$ on graded vector bundle $T{X}_0\oplus A,$ we prove the existence of a short exact sequence of diffeological groupoids over the discrete category $M$ (with appropriate vector space structures on the fibres) for the $\mbbX$ bundle $\ddot{I}€:P\to M.$ We introduce a notion of connection on $\mbbX$ bundle $\ddot{I}€:P\to M,$ and show that such a connection $\ddot{I}‰$ splits the sequence. Finally, we show that a connection pair $(\ddot{I}‰,\mathbb{H})$ on $\mbbX$ bundle $\ddot{I}€:P\to M$ is isomorphic to any other connection pair.}