  In this work, we show that a critical point of a 1d self-dual boundary phase
transition between two gapped boundaries of the $\mathbb{Z}_N$ topological
order can be described by a mathematical structure called an enriched fusion
category. The critical point of a boundary phase transition can be viewed as a
gappable non-chiral gapless boundary of the $\mathbb{Z}_N$ topological order. A
mathematical theory of the gapless boundaries of 2d topological orders
developed by Kong and Zheng (arXiv:1905.04924 and arXiv:1912.01760) tells us
that all macroscopic observables on the gapless boundary form an enriched
unitary fusion category, which can be obtained by a holographic principle
called the ``topological Wick rotation." Using this method, we obtain the
enriched fusion category that describes a critical point of the phase
transition between the $\mathbf{e}$-condensed boundary and the
$\mathbf{m}$-condensed boundary of the $\mathbb{Z}_N$ topological order. To
verify this idea, we also construct a lattice model to realize the critical
point and recover the mathematical data of this enriched fusion category. The
construction further shows that the categorical symmetry of the boundary is
determined by the topological defects in the bulk, which indicates the
holographic principle indirectly. This work shows, as a concrete example, that
the mathematical theory of the gapless boundaries of 2+1D topological orders is
a powerful tool to study general phase transitions.
