  We develop a systematic framework for understanding symmetries in topological
phases in 2+1 dimensions using the string-net model, encompassing both gauge
symmetries that preserve anyon species and global symmetries permuting anyon
species, including both invertible symmetries describable by groups and
noninvertible symmetries described by categories. As an archetypal example, we
reveal the first noninvertible categorical gauge symmetry of topological orders
in 2+1 dimensions: the Fibonacci gauge symmetry of the doubled Fibonacci
topological order, described by the Fibonacci fusion 2-category. Our approach
involves two steps: first, establishing duality between different string-net
models with Morita equivalent input fusion categories that describe the same
topological order; and second, constructing symmetry transformations within the
same string-net model when the dual models have isomorphic input fusion
categories, achieved by composing duality maps with isomorphisms of degrees of
freedom between the dual models.
