On some adjunctions in equivariant stable homotopy theory (Q1748446)

Sometimes, an exact functor \(f^*: T \longrightarrow S\) has a right adjoint \(f_*\), where \(f_*\) again has a right adjoint \(f^{(1)}\). This notion of a ``3-adjunction'' can be further generalised to a 5-, 6- and 7-adjunction. A naturally arising example of such a 3-adjunction comes from equivariant stable homotopy theory, where \(f^*\) is inflation, and \(f_*\) is the fixed point functor. The authors consider functors in equivariant stable homotopy theory, namely restriction, smashing with a finite spectrum, change of universe, inflation and geometric fixed points. They carefully examine the properties of these functors to show whether they fit into such a multiple adjunction, providing counterexamples where adjoints do not exist.