Correspondences and stable homotopy theory

A general method of producing correspondences and spectral categories out of symmetric ring objects in general categories is given. As an application, stable homotopy theory of spectra SH$SH$ is recovered from modules over a commutative symmetric ring spectrum defined in terms of framed correspondences over an algebraically closed field. Another application recovers stable motivic homotopy theory SH(k)$SH(k)$ from spectral modules over associated spectral categories.