  We develop a theory of cosupport and costratification in tensor triangular
geometry. We study the geometric relationship between support and cosupport,
provide a conceptual foundation for cosupport as categorically dual to support,
and discover surprising relations between the theory of costratification and
the theory of stratification. We prove that many categories in algebra,
topology and geometry are costratified by developing and applying descent
techniques. An overarching theme is that cosupport is relevant for diverse
questions in tensor triangular geometry and that a full understanding of a
category requires knowledge of both its support and its cosupport.
