In this talk I will discuss a constructible sheaf approach to (noncommutative) birational geometry of toric varieties through their derived categories. Coherent-constructible correspondence suggests one can associates to a toric GIT problem a natural stratification and linear families of polyhedral sheaves on a real torus. These sheaves provide a combinatorial model to study derived categories of simplicial toric varieties in a birational collection over the GKZ fan, and make it possible to compare the categories attached to different GIT chambers in the “total” Cox category. The emphasis will be on the relation between birational transformations and changes in the underlying polyhedral geometry.