2015/05/25 by Bojana Femić, Femić, Bojana
Mathematics · #FOS: Mathematics #Quantum Algebra (math.QA) #math.QA
paper · pdf · doi:10.48550/arxiv.1505.06504
Basically Section 5 is updated, making the construction of the cohomology groups more rigorous
arxiv created 2019/02/18 · arxiv updated 2019/02/19
We prove that if a finite tensor category \C is symmetric, then the monoidal category of one-sided \C-bimodule categories is symmetric. Consequently, the Picard group of \C (the subgroup of the Brauer-Picard group introduced by Etingov-Nikshych-Gelaki) is abelian in this case. We then introduce a cohomology over such \C. An important piece of tool for this construction is the computation of dual objects for bimodule categories and the fact that for invertible one-sided \C-bimodule categories the evaluation functor involved is an equivalence, being the coevaluation functor its quasi-inverse, as we show. Finally, we construct an infinite exact sequence a la Villamayor-Zelinsky for \C. It consists of the corresponding cohomology groups evaluated at three types of coefficients which repeat periodically in the sequence.