Geometric derived Hall algebra

We give a geometric formulation of Toen'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 Toen and Vezzosi in terms of model theoretical language.