# Characteristic class

Let $G$ be a topological group. A characteristic class of principal $G$-bundles is a natural transformation from the contravariant functor $b_G$ (which sends any topological space to the set of isomorphism classes of principal $G$-bundles on it) to the cohomology functor.
For a given topological space $X$, a characteristic class of principal $G$-bundles associates, to every principal $G$-bundle $P \to X$, an element $c(P) \in H^*(X)$, such that if $f:X \to Y$ is a continuous map, then $c(b_g^*(f)(P)) = H^*(f)(c(P))$.