  We investigate the duality structure of quantum lattice systems with
topological order, a collective order also appearing in fractional quantum Hall
systems. We define electromagnetic (EM) duality for all of Kitaev's quantum
double models based on discrete gauge theories with Abelian and non-Abelian
groups, and identify its natural habitat as a new class of topological models
based on Hopf algebras. We interpret these as extended string-net models,
whereupon Levin and Wen's string-nets, which describe all intrinsic topological
orders on the lattice with parity and time-reversal invariance, arise as
magnetic and electric projections of the extended models. We conjecture that
all string-net models can be extended in an analogous way, using more general
algebraic and tensor-categorical structures, such that EM duality continues to
hold. We also identify this EM duality with an invertible domain wall. Physical
applications include topology measurements in the form of pairs of dual tensor
networks.
