For $G$ a compactLie group there is a way to equip the topological classifying space$B G$ with smooth structure such that the corresponding smooth universal principal bundle$E G \to B G$ carries a smooth connection$\nabla_{univ}$ with the property that for every $G$-principal bundle$P \to X$ with connection$\nabla$ there is a smooth representative $f : X \to B G$ of the classifying map, such that $(P, \nabla) \simeq (P, f^* \nabla_{univ})$. This $\nabla_{univ}$ is called the universal $G$-connection.

Universal connections for manifolds of some bounded dimension $\leq n$ 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)

Mark Mostow, The differentiable space structures of Milnor classifying spaces, simplicial complexes, and geometric realizations, J. Differential Geom. Volume 14, Number 2 (1979), 255-293 (euclid:jdg/1214434974)

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