  Symmetry is a central concept for classical and quantum field theory,
usually, symmetry is described by a finite group or Lie group. In this work, we
introduce the weak Hopf algebra extension of symmetry, which arises naturally
in anyonic quantum systems; and we establish weak Hopf symmetry breaking theory
based on the fusion closed set of anyons. As a concrete example, we implement a
thorough investigation of the quantum double model based on a given weak Hopf
algebra and show that the vacuum sector of the model has weak Hopf symmetry.
The topological excitations and ribbon operators are discussed in detail. The
gapped boundary and domain wall theories are also established, we show that the
gapped boundary is algebraically determined by a comodule algebra, or
equivalently, a module algebra; and the gapped domain wall is determined by the
bicomodule algebra, or equivalently, a bimodule algebra. The microscopic
lattice constructions of the gapped boundary and domain wall are discussed in
detail. We also introduce the weak Hopf tensor network states, via which we
solve the weak Hopf quantum double lattice models on closed and open surfaces.
The duality of the quantum double phases is discussed in the last part.
