Fully exact and fully dualizable module categories
Azat M. Gainutdinov, Robert Laugwitz
TL;DR
This work introduces fully exact and perfect module categories over finite braided tensor categories as robust, non-semisimple analogues of separable module categories, and develops a comprehensive 2-categorical framework via relative Deligne products and internal algebras. It proves that fully exactness is stable under the relative Deligne product, and that perfect module categories are precisely the fully dualizable objects, providing a model for finite tensor 2-categories. The paper develops intrinsic characterizations through centralizer functors and braided opposites, studies duals and op-duals, and identifies when duals remain fully exact. It supplies nontrivial Hopf-algebra examples (including Sweedler and small quantum groups) showing both the boundaries and applications of these notions and outlines directions for idempotent completion, bimodule extensions, and center-based enhancements. Collectively, the results broaden Morita theory and dualizability in non-semisimple settings and offer a structured path toward higher 4D topological constructions using nonsemisimple tensor categories.
Abstract
We define fully exact module categories, a subclass of exact module categories over a finite braided tensor category that is stable under the relative Deligne product. In contrast, we demonstrate with examples in both zero and non-zero characteristic of the base field that the class of exact module categories is not stable under this product. We also observe in examples that fully exact module categories form a dense subset in the class of exact ones. The monoidal 2-category of fully exact module categories strictly contains those of invertible and separable module categories. In fact, we show that each internal algebra of a fully exact module category is projectively separable, a generalization of separable algebras involving projective objects. In the semisimple case, a module category is fully exact if and only if it is separable. In general, fully exact module categories are not dualizable inside their class, but if they are, they are fully dualizable objects in the monoidal 2-category of finite module categories. We call such module categories perfect. We show that perfect module categories form a rigid monoidal 2-subcategory containing all fully dualizable objects. Therefore, we propose perfect module categories as a model for finite tensor 2-categories. If the braiding is symmetric, a module category is fully exact if and only if it is perfect. As a detailed example, we classify fully exact, and hence perfect, module categories over the symmetric tensor category of modules over Sweedler's four-dimensional Hopf algebra and compute their relative Deligne products, and the categories of 1-morphisms. For a general quasi-triangular Hopf algebra, we analyze when the category of finite-dimensional vector spaces is fully exact. We show that this is not the case for both Sweedler's Hopf algebra and Lusztig's factorizable small quantum group of type $A_1$ at an odd root of unity.