2-Morita Theory of $E_2$-Algebras and Module Categories

Abstract: Building on our previous work on 2-Morita equivalence for $E_2$-algebras using topological pictures, we develop a systematic framework for Morita equivalence of topological orders in different dimensions in terms of $n$-Morita categories $\mathrm{Mrt}<em>{E_n}(\mathcal{C})$. In this framework, various notions of $n$-Morita equivalence are unified as equivalences of objects in $\mathrm{Mrt}</em>{E_n}(\mathcal{C})$. We also compare the constructions of higher Morita categories due to Haugseng and to Gwilliam--Scheimbauer. For $n=1,2$, we prove that the functor $$\mathrm{Mod}<em>n:\mathrm{Mrt}</em>{E_n}(\mathcal{C})\to \mathrm{Mrt}<em>{E</em>{n-1}}(\mathrm{LMod}<sup>{\mathrm{rep}}(\mathcal{C}))$$ is an equivalence, relating the algebraic description of higher Morita theory to its realization in terms of module categories. In this formulation, the notion of a bi-bimodule arises naturally and provides a common framework for local and confined modules. Its explicit orientation data, together with the corresponding fusion rules, clarifies the relations among defects arising from condensation in topological orders.