Nonabelian basechange theorems & \'etale homotopy theory

This paper has two main goals. First, we prove nonabelian refinements of basechange theorems in \'etale cohomology (i.e., prove analogues of the classical statements for sheaves of spaces). Second, we apply these theorems to prove a number of results about the \'etale homotopy type. Specifically, we prove nonabelian refinements of the smooth basechange theorem, Huber-Gabber affine analogue of the proper basechange theorem, and Fujiwara-Gabber rigidity theorem. Our methods also recover Chough's nonabelian refinement of the proper basechange theorem. Transporting an argument of Bhatt-Mathew to the nonabelian setting, we apply nonabelian proper basechange to show that the profinite \'etale homotopy type satisfies arc-descent. Using nonabelian smooth and proper basechange and descent, we give rather soft proofs of a number of K\"unneth formulas for the \'etale homotopy type.