In Müger's article "[Conformal Field Theory and Doplicher-Roberts Reconstruction](http://arxiv.org/abs/math-ph/0008027)", he defines the "modular closure" of a braided monoidal category. So every braided monoidal category (and therefore every premodular category, which is a special case, see Bruguières articles) is the subcategory of a modular category, namely of its closure.


Is it known when this inclusion is full? If not, is it possible for an arbitrary premodular category to construct a modular category that contains it as a full subcategory?



Is every premodular category the *full* subcategory of a modular category?
I believe the answer to your first question is no: If you take a symmetric tensor category, Müger's construction should give the trivial tensor category.


The answer to your second question is yes. There is always a full inclusion of a braided tensor category $\mathcal{C}$ into its center $\mathcal{Z}(\mathcal{C})$, which is modular if $\mathcal{C}$ is pre-modular.


