Non-Abelian Grothendieck Duality and Stable Homotopy

I will give a formulation of Grothendieck duality for derived algebraic
stacks. These are ordinary Artin/D-M stacks whose structure sheaf can be
enriched to a sheaf of multiplicative cohomology theories. In our
formulation, following Toen, Vessozi and Lurie, these objects are
$(\infty,1)$-topoi equipped with a structure sheaf of $E_{\infty}$-rings.
The category of quasi-coherent modules over such derived stacks form
symmetric monoidal stable model categories, which are natural homotopical
generalizations of abelian categories.

I will relate duality in this context to certain computational aspects of
stable homotopy theory. It is common to have generalized cohomology
theories arise as homotopy global sections of certain derived stacks.
Examples of these are the theory of topological modular forms, real
$K$-theory, real Morava E-theories and stable homotopy itself. Computing the
coefficient rings of these theories involve computing Ext in a category of
comodules over a Hopf algebroid, which is like a stacky version of group
cohomology. The associated ordinary Tate cohomology object can be
interpreted concretely in terms of derived duality.