Suppose $\mathcal{C}$ is a unitary ribbon fusion category. Also assume that its symmetric centre has trivial twist and trivial pivotal structure, i.e. is tannakian. Thus, the Müger/Bruguières modularisation/deequivariantisation exists.

Is the resulting modular fusion category unitary? Is the modularisation functor unitary (dagger)?

It feels to me like it should be obviously true, coming from the philosophy that the modularisation is a kind of generalised fibre functor. But I can't find a reference, nor can I write down the dagger structure.


1 Answer 1


The answer is yes to both questions (see Müger's paper). Müger's version of modularization is done in the unitary setting. The only new information that you need for full generality is that the unitary Drinfel'd center of a unitary fusion category is equal to the usual Drinfel'd center, (see [M])

More generally, a de-equivariantization of a unitary fusion category is again unitary (modularization corresponds to a specific kind of de-equivariantization). In fact, a $G$-de-equivariantization of a fusion category $\mathcal{C}$ is associated to a braided inclusion of Rep$(G)$ in $\mathcal{Z}(\mathcal{C})$. The de-equivariantization is the fusion category of left $\mathcal{O}_k(G)$-modules in $\mathcal{C}$. Now, if $\mathcal{C}$ is unitary, then the Drinfel'd center is a unitary modular category. Therefore, $\mathcal{O}_k(G)$ is a $Q$-system in $\mathcal{C}$. Thus, the de-equivariantization is unitary.


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