correct docs on TY MF example

Author: borisdevos

docs/src/man/multifusioncats.md

@@ -196,6 +196,6 @@ $$\mathcal{C} = \begin{pmatrix} \mathcal{C}_1 & \mathcal{M} \\ \mathcal{M}^{\tex
 
 We already identified the off-diagonal elements with module categories over the fusion categories on the diagonal. Accordingly, $\mathcal{M}$ is a $(\mathcal{C}_1, \mathcal{C}_2)$-bimodule category, and $\mathcal{M}^{\text{op}}$ is the opposite module category and a $(\mathcal{C}_2, \mathcal{C}_1)$-bimodule category. 
 
-If we take $\mathcal{C}_1 = \mathcal{C}_2 = \mathsf{Rep}(\mathbb{Z}_2)$ and $\mathcal{M} = \mathsf{Vec}$, then the entire multifusion category is isomorphic to the $\mathsf{Ising}$ category [etingof2016tensor; Example 4.10.5](@cite). We identify the trivial representation of $\mathsf{Rep}(\mathbb{Z}_2)$ with the unit of $\mathsf{Ising}$, the sign representation with $\psi$ and the unique object of $\mathsf{Vec}$ with the duality object $\sigma$. One can easily check that the fusion rules of $\mathsf{Ising}$ match with those we expect within $\mathsf{Rep}(\mathbb{Z}_2)$ and with its module category $\mathsf{Vec}$. Additionally, the fusion between $\mathsf{Vec}$ and $\mathsf{Vec}^\text{op}$ (and vice-versa) giving every object in $\mathcal{C}_1$ ($\mathcal{C}_2$) is consistent with $\sigma \times \sigma^* = 1 + \psi$. This particular example can be found in [TensorKitSectors](https://github.com/QuantumKitHub/TensorKitSectors.jl).
+If we take $\mathcal{C}_1 = \mathcal{C}_2 = \mathsf{Rep}(\mathbb{Z}_2)$ and $\mathcal{M} = \mathsf{Vec}$, then the entire multifusion category is Morita equivalent to $\mathsf{Rep}(\mathbb{Z}_2)$, and we view the $\mathbb{Z}_2$-extension of $\mathsf{Rep}(\mathbb{Z}_2)$ to be precisely the $\mathsf{Ising}$ category  [etingof2009; Section 9](@cite) [etingof2016tensor; Example 4.10.5](@cite). We identify the trivial representation of $\mathsf{Rep}(\mathbb{Z}_2)$ with the unit of $\mathsf{Ising}$, the sign representation with $\psi$ and the unique object of $\mathsf{Vec}$ with the duality object $\sigma$. One can easily check that the fusion rules of $\mathsf{Ising}$ match with those we expect within $\mathsf{Rep}(\mathbb{Z}_2)$ and with its module category $\mathsf{Vec}$. Additionally, the fusion between $\mathsf{Vec}$ and $\mathsf{Vec}^\text{op}$ (and vice-versa) giving every object in $\mathcal{C}_1$ ($\mathcal{C}_2$) is consistent with $\sigma \times \sigma^* = 1 + \psi$. This particular example can be found in [TensorKitSectors](https://github.com/QuantumKitHub/TensorKitSectors.jl).
 
-This construction can be generalised to $\mathcal{C}_1 = \mathcal{C}_2 = \mathsf{Rep(G)}$ with $\mathsf{G}$ a finite abelian group, such that the entire multifusion category is isomorphic to the Tambara-Yamagami category $\mathsf{TY}(\mathsf{G})$ (with positive Frobenius-Schur indicator for our purposes), and $\mathsf{Vec}$ will represent the duality object which squares to all invertible objects of the original group. To be exact, one of the diagonal fusion categories should be $\mathsf{Vec_G}$ for the correct Morita dual relation, but it is known for abelian groups that this is isomorphic to $\mathsf{Rep(G)}$.
+This construction can be generalised to $\mathcal{C}_1 = \mathcal{C}_2 = \mathsf{Rep(G)}$ with $\mathsf{G}$ a finite abelian group, such that the entire multifusion category is Morita equivalent to $\mathsf{Rep(G)}$ and can be evaluated as the Tambara-Yamagami category $\mathsf{TY}(\mathsf{G})$ (with positive Frobenius-Schur indicator for our purposes), and $\mathsf{Vec}$ will represent the duality object which squares to all invertible objects of the original group. To be exact, one of the diagonal fusion categories should be $\mathsf{Vec_G}$ for the correct Morita dual relation, but it is known for abelian groups that this is isomorphic to $\mathsf{Rep(G)}$.