A characteristic class c of principal G-bundles is then a natural transformation from bG to a cohomology functor H*, regarded also as a functor to Set.
In other words, a characteristic class associates to any principal G-bundle P → X an element c(P) in H*(X) such that, if f : Y → X is a continuous map, then c(f*P) = f*c(P). On the left is the class of the pullback of P to Y; on the right is the image of the class of P under the induced map in cohomology.
Read more about Characteristic Class: Characteristic Numbers, Motivation, Stability
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