Using the differentiable structure, twisted \(2\)periodic de Rham cohomology is well known and shows up as the target of Chern characters for twisted \(K\)theory. The main motivation of this work is a topological interpretation of twoperiodic twisted de Rham cohomology, which is generalizable to arbitrary topological spaces and at the same time to arbitrary coefficients. To this end the authors develop a sheaf theory in the context of locally compact topological stacks with emphasis on the construction of the sheaf theory operations in unbounded derived categories, elements of Verdier duality, and integration. The main result is the construction of a functorial periodization associated to a \(U(1)\)gerbe. As an application the authors verify the \(T\)duality isomorphism in periodic twisted cohomology and in periodic twisted orbispace cohomology. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Readership Graduate students and research mathematicians interested in geometry and topology. Table of Contents  Introduction
 Gerbes and periodization
 Functorial periodization
 \(T\)duality
 Orbispaces
 Verdier duality for locally compact stacks
 Bibliography
