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.
Universal connections for manifolds of some bounded dimension are appealed to in
Shiing-shen Chern, Differential geometry of fiber bundles, in: Proceedings of the International Congress of Mathematicians, Cambridge, Mass., (August-September 1950), vol. 2, pages 397-411, Amer. Math. Soc., Providence, R. I. (1952) (pdf, full proceedings vol 2 pdf)
(in the context of Chern-Weil theory)
and discussed in detail in
Mudumbai Narasimhan and Sundararaman Ramanan, Existence of Universal Connections, American Journal of Mathematics Vol. 83, No. 3 (Jul., 1961), pp. 563-572 (jstor:2372896)
Mudumbai Narasimhan and Sundararaman Ramanan, Existence of Universal Connections II, American Journal of Mathematics Vol. 85, No. 2 (Apr., 1963), pp. 223-231 (jstor:2373211)
Roger Schlafly, Universal connections, Invent Math 59, 59–65 (1980) (doi:10.1007/BF01390314)
Roger Schlafly, Universal connections: the local problem, Pacific J. Math. Volume 98, Number 1 (1982), 157-171 (euclid:pjm/1102734394)
See also
Discussion of universal connections on some smooth incarnation of the full classifying space:
Using diffeological spaces:
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