  We give a geometric formulation of To\"en's derived Hall algebra by
constructing Grothendieck's six operations for the derived category of
lisse-\'etale constructible sheaves on the derived stacks of complexes. Our
formulation is based on an variant of Laszlo and Olsson's theory of derived
categories and six operations for algebraic stacks. We also give an
$\infty$-theoretic explanation of the theory of derived stacks, which was
originally constructed by To\"en and Vezzosi in terms of model theoretical
language.
