By creating a dual functor of localization, co-localization functor, in a compactly generated tensor triangulated category \({\mathscr {T}},\) we develop a theory, cosupport, dual to that of support in \({\mathscr {T}}.\) We show that there are many properties that are similar or rather dual between cosupport and support in \({\mathscr {T}}.\) For instance, the cosupport of an object can be detected by the cosupport of its cohomology; the cosupport of a nonzero object is nonempty. Also some characterizations, properties and relations of cosupport and support are given.
