We prove equivalences of derived categories for the various mirrors in the Batyrev-Borisov construction. In particular, we obtain a positive answer to a conjecture of Batyrev and Nill. The proof involves passing to an associated category of singularities and toric variation of geometric invariant theory quotients.The second-named author thanks the Pacific Institute for the Mathematical Sciences and NSERC for various forms of support. Frequent visits to the University of Alberta greatly expedited the progress of this work. The first-named author is also indebted to the NSERC for support provided by a Canada Research Chair and Discovery Grant. The second-named author acknowledges that this material is based upon work supported by the National Science Foundation under Award No. DMS-1401446 and the Engineering and Physical Sciences Research Council under Grant EP/N004922/1.This is the author accepted manuscript. It is currently under an indefinite embargo pending publication by Johns Hopkins University Press
