  We develop the foundations of higher geometric stacks in complex analytic
geometry and in non-archimedean analytic geometry. We study coherent sheaves
and prove the analog of Grauert's theorem for derived direct images under
proper morphisms. We define analytification functors and prove the analog of
Serre's GAGA theorems for higher stacks. We use the language of infinity
category to simplify the theory. In particular, it enables us to circumvent the
functoriality problem of the lisse-\'etale sites for sheaves on stacks. Our
constructions and theorems cover the classical 1-stacks as a special case.
