  We investigate the multifusion generalization of string-net ground states and
lattice Hamiltonians, delving into its associated weak Hopf symmetry. For the
multifusion string-net, the gauge symmetry manifests as a general weak Hopf
algebra, leading to a reducible vacuum string label; the charge symmetry,
serving as a quantum double of gauge symmetry, constitutes a connected weak
Hopf algebra. This implies that the associated topological phase retains its
characterization by a unitary modular tensor category (UMTC). The bulk charge
symmetry can also be captured by a weak Hopf tube algebra. We offer an explicit
construction of the weak Hopf tube algebra structure and thoroughly discuss its
properties. The gapped boundary and domain wall models are extensively
discussed, with these $1d$ phases characterized by unitary multifusion
categories (UMFCs). We delve into the gauge and charge symmetries of these $1d$
phases, as well as the construction of the boundary and domain wall tube
algebras. Additionally, we illustrate that the domain wall tube algebra can be
regarded as a cross product of two boundary tube algebras. As an application of
our model, we elucidate how to interpret the defective string-net as a
restricted multifusion string-net.
