Given an (irreducible) Möbius covariant net \({\mathcal {A}}\), we prove a Bisognano–Wichmann theorem for its categorical extension \({\mathscr {E}}^{\mathrm d}\) associated with the braided \(C^*\)-tensor category \(\mathrm {Rep}^{\mathrm d}({\mathcal {A}})\) of dualizable (more precisely, “dualized”) Möbius covariant \({\mathcal {A}}\)-modules. As a closely related result, we prove a (modified) Bisognano–Wichmann theorem for any (possibly) non-local extension of \({\mathcal {A}}\) obtained by a \(C^*\)-Frobenius algebra Q in \(\mathrm {Rep}^{\mathrm d}({\mathcal {A}})\). As an application, we discuss the relation between the domains of modular operators and the preclosedness of certain unbounded operators in \({\mathscr {E}}^{\mathrm d}\).

给定一个（不可约的）莫比乌斯协变网络\({\mathcal {A}}\)，我们证明了 Bisognano–Wichmann 定理的分类扩展\({\mathscr {E}}^{\mathrm d}\)与编织的\(C^*\) -张量类别\(\mathrm {Rep}^{\mathrm d}({\mathcal {A}})\)的可二元化（更准确地说，“二元化”）莫比乌斯协变\( {\mathcal {A}}\) -模块。作为密切相关的结果，我们证明了通过\(C^*\) -Frobenius 代数获得的\({\mathcal {A}}\) 的任何（可能）非局部扩展的（修改的）Bisognano-Wichmann 定理Q in \(\mathrm {Rep}^{\mathrm d}({\mathcal {A}})\). 作为一个应用，我们讨论了模运算符的域与\({\mathscr {E}}^{\mathrm d}\)中某些无界运算符的预封闭性之间的关系。