In derived category, there is a slogan, "cohomology is bad, chain complex is good". In the theory of characteristic classes, we could associate a vector bundle to cohomology classes of the base space.

Can we lift up (naturally) these cohomology classes to an actual complex?

I am particularly interested in Chern class and vector bundle over an algebraic variety over complex numbers.

groupsthat are bad. Cohomology classes are fine (in fact they can be interpreted as maps between complexes) $\endgroup$ – Denis Nardin Jan 28 at 20:25