Pontrjagin duality on multiplicative gerbes

We use Segal–Mitchison’s cohomology of topological groups to define a convenient model for topological gerbes. We introduce multiplicative gerbes over topological groups in this setup and define their representations. For a specific choice of representation, we construct its category of endomorphisms, and we show that it induces a new multiplicative gerbe over another topological group. This new induced group is fiberwise Pontrjagin dual of the original one, and therefore we called the pair of multiplicative gerbes “Pontrjagin dual”. We show that Pontrjagin dual multiplicative gerbes have equivalent categories of representations. In addition, we show that their monoidal centers are equivalent. Examples of Pontrjagin dual multiplicative gerbes over finite and discrete, as well as compact and non-compact, Lie groups are provided.