INTERNATIONAL MATHEMATICS RESEARCH NOTICES, v.2021, no.10, pp.7799 - 7849
OXFORD UNIV PRESS
The main goal of this paper is to set a foundation for homotopy theory of algebraic stacks under model category theory and to show how it can be applied in various contexts. It not only generalizes the etale homotopy theory to algebraic stacks but also provides more suitable framework for the homotopy theory in a broader context. Also, a new result that the profinite completion of pro-simplicial sets admits a right adjoint is provided and integrated with the foundational work to generalize Artin-Mazur's comparison theorem from schemes to algebraic stacks in a formal way.