  We prove that extension groups in strict polynomial functor categories
compute the rational cohomology of classical algebraic groups. This result was
previously known only for general linear groups. We give several applications
to the study of classical algebraic groups, such as a cohomological
stabilization property, the injectivity of external cup products, and the
existence of Hopf algebra structures on the (stable) cohomology of a classical
algebraic group with coefficients in a Hopf algebra. Our result also opens the
way to new explicit cohomology computations. We give an example inspired by
recent computations of Djament and Vespa.
