Fiber product of condensable algebras and critical points of boundary phase transitions

In this work, we propose a mathematical description of the critical point of boundary phase transitions in 2+1D topological orders. Given two boundary phases, $\mathcal{C}_A$ and $\mathcal{C}_B$, with corresponding Lagrangian algebras $A$ and $B$ in a 2+1D topological order $\mathcal{C}$, respective