Cartier duality for gerbes of vector bundles

Using the theory of abstract six functor formalisms of
analytic stacks, I will explain a general Cartier duality for gerbes
of vector bundles in different algebraic and analytic setups. As an
application, admitting some foundational aspects of the theory of the
analytic Hodge-Tate stack (joint with Anschütz, Le Bras and Scholze),
one can deduce a Cartier duality between the categories of
quasi-coherent sheaves of the analytic
Hodge-Tate stack and weight one modules on the Simpson gerbe of Bhatt-Zhang.