A new mixed cochain complex for module categories is isomorphic to the Davydov-Yetter complex of the representation functor, and for finite cases its cohomology equals relative Ext groups Ext^•_{Z(C),C}(1, A_M) giving dimension formulas and H^{>0}=0.
Shimizu, Further results on the structure of (co)ends in finite tensor categories , Appl
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Deformations of mixed associators in module categories
A new mixed cochain complex for module categories is isomorphic to the Davydov-Yetter complex of the representation functor, and for finite cases its cohomology equals relative Ext groups Ext^•_{Z(C),C}(1, A_M) giving dimension formulas and H^{>0}=0.