The Compactness Locus of a Geometric Functor and the Formal Construction of the Adams Isomorphism

We introduce the compactness locus of a geometric functor between rigidly-compactly generated tensor-triangulated categories, and describe it for several examples arising in equivariant homotopy theory and algebraic geometry. It is a subset of the tensor-triangular spectrum of the target category which, crudely speaking, measures the failure of the functor to satisfy Grothendieck-Neeman duality (or equivalently, to admit a left adjoint). We prove that any geometric functor — even one which does not admit a left adjoint — gives rise to a Wirthmüller isomorphism once one passes to a colocalization of the target category determined by the compactness locus. When applied to the inflation functor in equivariant stable homotopy theory, this produces the Adams isomorphism.