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In Müger's article "Conformal Field Theory and Doplicher-Roberts Reconstruction", 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?

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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.

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  • $\begingroup$ How does this inclusion work? What is the canonical choice of half braiding $\phi: - \otimes X \To X \otimes -$ for a given object $X$? But only the morphisms that commute with the half braiding appear in the Drinfeld centre, right? So how is this an inclusion? $\endgroup$
    – Manuel Bärenz
    Commented Oct 29, 2013 at 18:47
  • $\begingroup$ Ah, if one takes the braiding present in $\mathcal{C}$ then all the morphisms in $\mathcal{C}$ satisfy the condition, by the axioms of a braided category. Thanks! $\endgroup$
    – Manuel Bärenz
    Commented Oct 29, 2013 at 19:04

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