Proper base change for étale sheaves of spaces.

The goal of this paper is to prove the proper base change theorem for truncated coherent étale sheaves of spaces, generalizing the usual proper base change in étale cohomology. As applications, we show that the profinite étale topological type functor commutes with finite products and the symmetric powers of proper algebraic spaces over a separably closed field, respectively. In particular, the commutativity of the étale fundamental groups with finite products is extended to all higher homotopy groups. [ABSTRACT FROM AUTHOR]