  We sketch a procedure to capture general non-invertible symmetries of a
d-dimensional quantum field theory in the data of a higher-category, which
captures the local properties of topological defects associated to the
symmetries. We also discuss fusions of topological defects, which involve
condensations/gaugings of higher-categorical symmetries localized on the
worldvolumes of topological defects. Recently some fusions of topological
defects were discussed in the literature where the dimension of topological
defects seems to jump under fusion. This is not possible in the standard
description of higher-categories. We explain that the dimension-changing
fusions are understood as higher-morphisms of the higher-category describing
the symmetry. We also discuss how a 0-form sub-symmetry of a higher-categorical
symmetry can be gauged and describe the higher-categorical symmetry of the
theory obtained after gauging. This provides a procedure for constructing
non-invertible higher-categorical symmetries starting from invertible
higher-form or higher-group symmetries and gauging a 0-form symmetry. We
illustrate this procedure by constructing non-invertible 2-categorical
symmetries in 4d gauge theories and non-invertible 3-categorical symmetries in
5d and 6d theories. We check some of the results obtained using our approach
against the results obtained using a recently proposed approach based on 't
Hooft anomalies.
