Connections on a principal Lie groupoid bundle and representations up to homotopy
Saikat Chatterjee, Naga Arjun S J
TL;DR
This work extends classical principal bundle connections to principal Lie groupoid bundles by employing an Ehresmann connection on the groupoid to define actions up to homotopy on graded bundles $A \oplus TX_0$. It constructs three diffeological groupoids—$\mathcal{A}\mathfrak{d}(P)$, $\mathcal{A}\mathfrak{t}(P)$, and $\mathcal{A}\mathfrak{ct}(\pi^*TM)$—and a fundamental vector-field functor $\bar{d}\delta$ forming an Atiyah sequence for the $\mathbb{X}$-bundle $P\to M$, with a connection providing a splitting. When the Lie groupoid connection is Cartan, the framework yields a full equivalence between connections and representations up to homotopy, and Cartan-compatibility ensures exactness of the Atiyah sequence in the diffeological sense. The theory paves the way for applications to differentiable stacks and foliation geometry, and establishes a solid bridge between higher gauge concepts and diffeological structures.
Abstract
A Lie groupoid principal $\mbbX$ bundle is a surjective submersion $π\colon P\to M$ 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 $A:=1^*\ker ds\to X_0$, 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 $\mathbb{H}\subset TX_1$ on the Lie groupoid $\mathbb{X},$ and considering its induced action up to homotopy of $\mathbb{X}$ on graded vector bundle $TX_0\oplus A,$ 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 $π\colon P\to M.$ We introduce a notion of connection on $\mbbX$ bundle $π\colon P\to M,$ and show that such a connection $ω$ splits the sequence. Finally, we show that a connection pair $(ω, \mathbb{H})$ on $\mbbX$ bundle $π\colon P\to M$ is isomorphic to any other connection pair.}