For a compact Lie group there is a way to equip the topological classifying space with smooth structure such that the corresponding smooth universal principal bundle carries a smooth connection with the property that for every -principal bundle with connection there is a smooth representative of the classifying map, such that . This is called the universal -connection.
M. S. Narasimhan and S. Ramanan,
Existence of universal connections , Amer. J. Math. 83 (1961), 563–572. MR 24 #A3597
Existence of universal connections II , Amer. J. Math. 85 (1963), 223–231. MR 27 #1904