Let Formula: see text be a commutative algebra in a braided monoidal category Formula: see text. For example, Formula: see text could be a vertex operator algebra (VOA) extension of a VOA Formula: see text in a category Formula: see text of Formula: see text-modules. We first find conditions for the category Formula: see text of Formula: see text-modules in Formula: see text and its subcategory Formula: see text of local modules to inherit rigidity from Formula: see text. Second and more importantly, we prove a converse result, finding conditions under which Formula: see text and Formula: see text inherit rigidity from Formula: see text. For our first results, we assume that Formula: see text is a braided finite tensor category and identify mild conditions under which Formula: see text and Formula: see text are also rigid. These conditions are based on criteria due to Etingof and Ostrik for Formula: see text to be an exact algebra in Formula: see text. As an application, we show that if Formula: see text is a simple Formula: see text-graded VOA containing a strongly rational vertex operator subalgebra Formula: see text, then Formula: see text is also strongly rational, without requiring the dimension of Formula: see text in the modular tensor category of Formula: see text-modules to be non-zero. We also identify conditions under which the category of Formula: see text-modules inherits rigidity from the module category of a Formula: see text-cofinite non-rational subalgebra Formula: see text. For our converse result, we assume that Formula: see text is a Grothendieck–Verdier category, which means that Formula: see text admits a weaker duality structure than rigidity. We first show that Formula: see text is also a Grothendieck–Verdier category. Using this, we then prove that if Formula: see text is rigid, then so is Formula: see text under conditions that include a mild non-degeneracy assumption on Formula: see text, as well as assumptions that every simple object of Formula: see text is local and that induction Formula: see text commutes with duality. These conditions are motivated by free field-like VOA extensions Formula: see text where Formula: see text is often an indecomposable Formula: see text-module, and thus our result will make it more feasible to prove rigidity for many vertex algebraic braided monoidal categories. In a follow-up work, our results are used to prove rigidity of the category of weight modules for the simple affine VOA of Formula: see text at any admissible level, which embeds by Adamović’s inverse quantum Hamiltonian reduction into a rational Virasoro VOA tensored with a half-lattice VOA.
Creutzig et al. (Sun,) studied this question.