  Generalized symmetry, including non-invertible and categorical symmetry,
plays a central role in contemporary studies on topological orders (TOs) and
the corresponding conformal field theories (CFTs). The generators of such
symmetries have a close connection to non-abelian anyonic objects in a bulk CFT
or chiral CFT (CCFT), but it has been known that the construction of a CCFT
contains theoretical difficulties in general. In this work, we revisit the
structure of the fusion rule in $Z_{N}$ symmetric chiral and bulk conformal
field theories and the corresponding TOs. We propose a nontrivial expression of
subalgebra structure in the fusion rule of a bulk CFT. We name this subalgebra
``bulk semion" which corresponds to the fusion rule of the CCFTs and
categorical symmetry of the TOs. This is a bulk-edge correspondence based on
the symmetry analysis and can be interpreted as a version of topological
holography in the recent literature. The topological holography has been
expected to be applicable to the systems in general space-time dimensions.
Moreover, we give a concise way of unifying duality (or fractional
supersymmetry), generalized or categorical symmetry, and Lagrangian subalgebra.
Our method is potentially useful to formulate and study general TOs,
fundamentally only from the data of bulk CFTs or vice versa, and gives a clue
in understanding CCFT (or ancillary CFT more generally).
