Factorization for stacks and boundary complexes

We prove a weak factorization result on birational maps of Deligne-Mumford stacks, and deduce the following: Let $U\subset X$ be an open embedding of smooth Deligne-Mumford stacks such that $D=X-U$ is a normal crossings divisor, then the the simple homotopy type of the boundary complex $\mathrm{\Delta }(X,D)$ depends only on $U$.