Hochschild–Mitchell cohomology and Galois extensions

We define HH-Galois extensions for kk-linear categories and prove the existence of a Grothendieck spectral sequence for Hochschild–Mitchell cohomology related to this situation. This spectral sequence is multiplicative and for a group algebra decomposes as a direct sum indexed by conjugacy classes of the group. We also compute some Hochschild–Mitchell cohomology groups of categories with infinite associated quivers.