Rigidity of non-negligible objects of moderate growth in braided categories

Let k be a field, and let C be a Cauchy complete k -linear braided category with finite-dimensional morphism spaces and . We call an indecomposable object X of C non-negligible if there exists Y∈ C such that is a direct summand of Y⊗ X . We prove that every non-negligible object X∈ C such that End(X⊗ n)<n! for some n is automatically rigid. In particular, if C is semisimple of moderate growth and weakly rigid, then C is rigid. As applications, we simplify Huang’s proof of rigidity of representation categories of certain vertex operator algebras, and we get that for a finite semisimple monoidal category C , the data of a C -modular functor is equivalent to a modular fusion category structure on C , answering a question of Bakalov and Kirillov. Furthermore, we show that if C is rigid and has moderate growth, then the quantum trace of any nilpotent endomorphism in C is zero. Hence C admits a semisimplification, which is a semisimple braided tensor category of moderate growth. Finally, we discuss rigidity in braided r-categories which are not semisimple, which arise in logarithmic conformal field theory. These results allow us to simplify a number of arguments of Kazhdan and Lusztig.