The Modular Closure of Braided Tensor Categories

Abstract

With every monoid M one can associate its center Z(M)={x∈M|xy=yx∀y∂M}, which obviously is a commutative monoid. If M has inverses, i.e. is a group, then Z(M) is a normal subgroup and one can consider the quotient group M|Z(M). In nice cases, e.g. if M is a direct product of simple groups, M|Z(M) turns out to have trivial center. Monoids being 0-categories, the work to be reported here can be considered as the analogous construction for 1-categories. We refer to Müger (1998) for a full account. Given a strict tensor category ∋ with braiding ε, we define its center to be the full subcategory defined by

$$ ObjZ\left( \mathcal{C} \right) = \left\{ {\rho \in Obj\mathcal{C}\left| {\varepsilon \left( {\rho ,\sigma } \right)o\varepsilon \left( {\sigma ,\rho } \right) = id_{\sigma \otimes \rho } \forall \sigma \in Obj\mathcal{C}} \right.} \right\}, $$

, which clearly is a symmetric tensor category.