  We investigate a version of the Green correspondence for categories of
complexes, including homotopy categories and derived categories. The
correspondence is an equivalence between a category defined over a finite group
$G$ and the same for a subgroup $H$, often the normalizer of a $p$-subgroup of
$G$. We present a basic formula for deciding when categories of modules or
complexes have a Green correspondence and apply it to many examples. In several
cases the equivalence is an equivalence of triangulated categories, and in
special cases it is an equivalence of tensor triangulated categories.
