The Morita Theory of Fusion 2-Categories
Thibault D. Décoppet
TL;DR
This work elevates Morita theory to fusion 2-categories by categorifying the classical notions through separable algebras, separable module 2-categories, and dual tensor 2-categories. It establishes the relative tensor product over separable algebras, constructs the Morita 3-category, and proves a 2-sided Eilenberg–Watts style correspondence between algebras, modules, and endomorphisms. The paper then defines dual tensor 2-categories and proves three equivalent characterizations of Morita equivalence for locally separable tensor 2-categories, linking separable algebras to module categories and their endomorphism 2-categories. It further connects these higher-categorical Morita notions to familiar 1-categorical structures, recovering results such as Witt equivalence and providing tools to generate new fusion 2-categories via tensor duals. Overall, the results formalize Morita theory in the fusion 2-category setting and lay groundwork for broader 4-category duality conjectures and internal Morita theory.
Abstract
We develop the Morita theory of fusion 2-categories. In order to do so, we begin by proving that the relative tensor product of modules over a separable algebra in a fusion 2-category exists. We use this result to construct the Morita 3-category of separable algebras in a fusion 2-category. Then, we go on to explain how module 2-categories form a 3-category. After that, we define separable module 2-categories over a fusion 2-category, and prove that the Morita 3-category of separable algebras is equivalent to the 3-category of separable module 2-categories. As a consequence, we show that the dual tensor 2-category with respect to a separable module 2-category, that is the associated 2-category of module 2-endofunctors, is a multifusion 2-category. Finally, we give three equivalent characterizations of Morita equivalence between fusion 2-categories.