We clarify the relationship between Grothendieck dualityà la Neeman and the Wirthmüller isomorphismà la Fausk-Hu-May. We exhibit an interesting pattern of symmetry in the existence of adjoint functors between compactly generated tensor-triangulated categories, which leads to a surprising trichotomy: There exist either exactly three adjoints, exactly five, or infinitely many. We highlight the importance of so-called relative dualizing objects and explain how they give rise to dualities on canonical subcategories. This yields a duality theory rich enough to capture the main features of Grothendieck duality in algebraic geometry, of generalized Pontryagin-Matlis dualityà la Dwyer-Greenless-Iyengar in the theory of ring spectra, and of Brown-Comenetz dualityà la Neeman in stable homotopy theory.
We show that the bicategory of finite groupoids and right-free permutation bimodules is a quotient of the bicategory of Mackey 2-motives introduced in [BD20], obtained by modding out the so-called cohomological relations. This categorifies Yoshida's Theorem for ordinary cohomological Mackey functors, and provides a direct connection between Mackey 2-motives and the usual blocks of representation theory.
Abstract. For equivariant stable homotopy theory, equivariant KK-theory and equivariant derived categories, we show how restriction to a subgroup of finite index yields a finite commutative separable extension, analogous to finité etale extensions in algebraic geometry.
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