Universal connections on Lie groupoids (Q1811919)

There are several different notions of universal connections for connections in principal fibre bundles. One approach was suggested by \textit{M. S. Narasimhan} and \textit{S. Ramanan} [Am. J. Math. 83, 563-572 (1961; Zbl 0114.38203) and ibid. 85, 223-231 (1963; Zbl 0117.39002)]. The authors base their study on another approach proposed by \textit{P. L. Garcia} [Rend. Sem. Mat. Univ. Padova 47, 227-242 (1972; Zbl 0251.53024)]. For a given Lie groupoid \(\Pi\) they construct another Lie groupoid with a connection \(\omega\) in it such that any connection in \(\Pi\) is obtained as a pull-back of \(\omega\) via a suitably chosen map. In the last section of the paper they explain how to recover the result of Garcia from their theorem. Moreover, they show that in the case of the frame groupoid \(\Pi(E)\) of a vector bundle \(E\) they obtain a linear universal connection for \(E\) in some other vector bundle \(E_1\).