  In this paper we discuss a general notion of Weil cohomology theories, both
in algebraic geometry and in rigid analytic geometry. We allow our Weil
cohomology theories to have coefficients in arbitrary commutative ring spectra.
Using the theory of motives, we give three equivalent viewpoints on Weil
cohomology theories: as a cohomology theory on smooth varieties, as a motivic
spectrum and as a realization functor. We also associate to every Weil
cohomology theory a motivic Hopf algebroid generalizing a previous construction
of the author for the Betti cohomology. The main result proven in the paper is
the connectivity of the motivic Hopf algebroids associated to the classical
Weil cohomology theories.
