On Deformations of Module Categories over Finite Tensor Categories

Abstract

In this thesis, we introduce and study infinitesimal associator deformations of a module category M over a monoidal category C, including deformations of non-strict module endofunctors. We construct a cochain complex (allowing module endofunctors as coeffi cients), called the associator complex, whose cohomology controls these deformations up to equivalence. In particular, its second cohomology with trivial coefficients describes infinitesimal deformations of mixed associators of M while the first cohomology classifies infinitesimal deformations of module structures of a C-module functor. We show that the associator complex admits the structure of a dg-algebra and a dg-module over the Davydov-Yetter complex of the identity functor of the monoidal category C. We establish an isomorphism between the associator complex and the Davydov-Yetter complex of the action functor, which also respects the dg-algebra and dg-module structure. In the case, where C is a finite tensor category, we use techniques from relative homological algebra to realize the asssociator cohomology of a finite module category M (with coefficients) as a relative Ext for the standard adjunction between the categories of module endofunctors and linear endofunctors of M. In the case where M is an exact C-module category, we show furthermore that the associator cohomology with coefficients is isomorphic to the relative Ext of a much simpler adjunction, the one between the Drinfeld center Z(C) and C. In particular, for trivial coefficients, the total associator cohomology is the total relative Ext group between the tensor unit of C and the adjoint algebra of M. This is the main result of the thesis, which is based on another important technical result for liftings of C-bimodule functors. We use this result first to show that the regular module category never admits associator deformations. We also apply it in explicit calculations for the case of Hopf algebras, and find non-trivial associator deformations of module categories over Sweedler's 4-dimensional Hopf algebra, including its generalizations.